feat(library/theories/group_theory/*): add new development of group theory
This commit is contained in:
Jeremy Avigad
Leonardo de Moura
parent
f8a8502b14
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e6fd644526
7 changed files with 1869 additions and 4 deletions
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theories.group_theory
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=====================
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theories.finite_group_theory
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============================
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Group theory (especially finite group theory).
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973
library/theories/group_theory/basic.lean
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973
library/theories/group_theory/basic.lean
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/-
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Copyright (c) 2016 Andrew Zipperer. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Andrew Zipperer, Jeremy Avigad
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Basic group theory: subgroups, homomorphisms on a set, homomorphic images, cosets,
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normal cosets and the normalizer, the kernel of a homomorphism, the centralizer, etc.
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For notation a * S and S * a for cosets, open the namespace "coset_notation".
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For notation a^b and S^a, open the namespace "conj_notation".
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TODO: homomorphisms on sets should be refactored and moved to algebra.
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-/
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import data.set algebra.homomorphism theories.topology.basic theories.move
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open eq.ops set function
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namespace group_theory
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variables {A B C : Type}
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/- subgroups -/
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structure is_one_closed [class] [has_one A] (S : set A) : Prop :=
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(one_mem : one ∈ S)
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proposition one_mem [has_one A] {S : set A} [is_one_closed S] : 1 ∈ S :=
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is_one_closed.one_mem _ S
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structure is_mul_closed [class] [has_mul A] (S : set A) : Prop :=
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(mul_mem : ∀₀ a ∈ S, ∀₀ b ∈ S, a * b ∈ S)
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proposition mul_mem [has_mul A] {S : set A} [is_mul_closed S] {a b : A} (aS : a ∈ S) (bS : b ∈ S) :
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a * b ∈ S :=
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is_mul_closed.mul_mem _ S aS bS
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structure is_inv_closed [class] [has_inv A] (S : set A) : Prop :=
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(inv_mem : ∀₀ a ∈ S, a⁻¹ ∈ S)
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proposition inv_mem [has_inv A] {S : set A} [is_inv_closed S] {a : A} (aS : a ∈ S) : a⁻¹ ∈ S :=
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is_inv_closed.inv_mem _ S aS
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structure is_subgroup [class] [group A] (S : set A)
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extends is_one_closed S, is_mul_closed S, is_inv_closed S : Prop
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section groupA
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variable [group A]
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proposition mem_of_inv_mem {a : A} {S : set A} [is_subgroup S] (H : a⁻¹ ∈ S) : a ∈ S :=
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have (a⁻¹)⁻¹ ∈ S, from inv_mem H,
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by rewrite inv_inv at this; apply this
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proposition inv_mem_iff (a : A) (S : set A) [is_subgroup S] : a⁻¹ ∈ S ↔ a ∈ S :=
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iff.intro mem_of_inv_mem inv_mem
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proposition is_subgroup_univ [instance] : is_subgroup (@univ A) :=
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⦃ is_subgroup,
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one_mem := trivial,
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mul_mem := λ a au b bu, trivial,
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inv_mem := λ a au, trivial ⦄
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proposition is_subgroup_inter [instance] (G H : set A) [is_subgroup G] [is_subgroup H] :
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is_subgroup (G ∩ H) :=
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⦃ is_subgroup,
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one_mem := and.intro one_mem one_mem,
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mul_mem := λ a ai b bi, and.intro (mul_mem (and.left ai) (and.left bi))
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(mul_mem (and.right ai) (and.right bi)),
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inv_mem := λ a ai, and.intro (inv_mem (and.left ai)) (inv_mem (and.right ai)) ⦄
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end groupA
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/- homomorphisms on sets -/
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section has_mulABC
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variables [has_mul A] [has_mul B] [has_mul C]
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-- in group theory, we can use is_hom for is_mul_hom
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abbreviation is_hom := @is_mul_hom
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definition is_hom_on [class] (f : A → B) (S : set A) : Prop :=
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∀₀ a₁ ∈ S, ∀₀ a₂ ∈ S, f (a₁ * a₂) = f a₁ * f a₂
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proposition hom_on_mul (f : A → B) {S : set A} [H : is_hom_on f S] {a₁ a₂ : A}
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(a₁S : a₁ ∈ S) (a₂S : a₂ ∈ S) : f (a₁ * a₂) = (f a₁) * (f a₂) :=
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H a₁S a₂S
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proposition is_hom_on_of_is_hom (f : A → B) (S : set A) [H : is_hom f] : is_hom_on f S :=
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forallb_of_forall₂ S S (hom_mul f)
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proposition is_hom_of_is_hom_on_univ (f : A → B) [H : is_hom_on f univ] : is_hom f :=
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is_mul_hom.mk (forall_of_forallb_univ₂ H)
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proposition is_hom_on_univ_iff (f : A → B) : is_hom_on f univ ↔ is_hom f :=
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iff.intro (λH, is_hom_of_is_hom_on_univ f) (λ H, is_hom_on_of_is_hom f univ)
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proposition is_hom_on_of_subset (f : A → B) {S T : set A} (ssubt : S ⊆ T) [H : is_hom_on f T] :
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is_hom_on f S :=
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forallb_of_subset₂ ssubt ssubt H
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proposition is_hom_on_id (S : set A) : is_hom_on id S :=
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have H : is_hom (@id A), from is_mul_hom_id,
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is_hom_on_of_is_hom id S
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proposition is_hom_on_comp {S : set A} {T : set B} {g : B → C} {f : A → B}
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(H₁ : is_hom_on f S) (H₂ : is_hom_on g T) (H₃ : maps_to f S T) : is_hom_on (g ∘ f) S :=
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take a₁, assume a₁S, take a₂, assume a₂S,
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have f a₁ ∈ T, from H₃ a₁S,
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have f a₂ ∈ T, from H₃ a₂S,
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show g (f (a₁ * a₂)) = g (f a₁) * g (f a₂), by rewrite [H₁ a₁S a₂S, H₂ `f a₁ ∈ T` `f a₂ ∈ T`]
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end has_mulABC
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section groupAB
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variables [group A] [group B]
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proposition hom_on_one (f : A → B) (G : set A) [is_subgroup G] [H : is_hom_on f G] : f 1 = 1 :=
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have f 1 * f 1 = f 1 * 1, by rewrite [-H one_mem one_mem, *mul_one],
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eq_of_mul_eq_mul_left' this
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proposition hom_on_inv (f : A → B) {G : set A} [is_subgroup G] [H : is_hom_on f G]
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{a : A} (aG : a ∈ G) :
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f a⁻¹ = (f a)⁻¹ :=
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have f a⁻¹ * f a = 1, by rewrite [-H (inv_mem aG) aG, mul.left_inv, hom_on_one f G],
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eq_inv_of_mul_eq_one this
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proposition is_subgroup_image [instance] (f : A → B) (G : set A)
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[is_subgroup G] [is_hom_on f G] :
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is_subgroup (f ' G) :=
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⦃ is_subgroup,
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one_mem := mem_image one_mem (hom_on_one f G),
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mul_mem := λ a afG b bfG,
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obtain c (cG : c ∈ G)(Hc : f c = a), from afG,
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obtain d (dG : d ∈ G)(Hd : f d = b), from bfG,
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show a * b ∈ f ' G, from mem_image (mul_mem cG dG) (by rewrite [hom_on_mul f cG dG, Hc, Hd]),
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inv_mem := λ a afG,
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obtain c (cG : c ∈ G)(Hc : f c = a), from afG,
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show a⁻¹ ∈ f ' G, from mem_image (inv_mem cG) (by rewrite [hom_on_inv f cG, Hc]) ⦄
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end groupAB
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/- cosets -/
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definition lcoset [has_mul A] (a : A) (N : set A) : set A := (mul a) 'N
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definition rcoset [has_mul A] (N : set A) (a : A) : set A := (mul^~ a) 'N
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-- overload multiplication
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namespace coset_notation
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infix * := lcoset
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infix * := rcoset
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end coset_notation
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open coset_notation
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section has_mulA
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variable [has_mul A]
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proposition mul_mem_lcoset {S : set A} {x : A} (a : A) (xS : x ∈ S) : a * x ∈ a * S :=
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mem_image_of_mem (mul a) xS
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proposition mul_mem_rcoset [has_mul A] {S : set A} {x : A} (xS : x ∈ S) (a : A) :
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x * a ∈ S * a :=
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mem_image_of_mem (mul^~ a) xS
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definition lcoset_equiv (S : set A) (a b : A) : Prop := a * S = b * S
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proposition equivalence_lcoset_equiv (S : set A) : equivalence (lcoset_equiv S) :=
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mk_equivalence (lcoset_equiv S) (λ a, rfl) (λ a b, !eq.symm) (λ a b c, !eq.trans)
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proposition lcoset_subset_lcoset {S T : set A} (a : A) (H : S ⊆ T) : a * S ⊆ a * T :=
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image_subset _ H
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proposition rcoset_subset_rcoset {S T : set A} (H : S ⊆ T) (a : A) : S * a ⊆ T * a :=
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image_subset _ H
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proposition image_lcoset_of_is_hom_on {B : Type} [has_mul B] {f : A → B} {S : set A} {a : A}
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{G : set A} (SsubG : S ⊆ G) (aG : a ∈ G) [is_hom_on f G] :
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f ' (a * S) = f a * f ' S :=
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ext (take x, iff.intro
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(assume fas : x ∈ f ' (a * S),
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obtain t [s (sS : s ∈ S) (seq : a * s = t)] (teq : f t = x), from fas,
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have x = f a * f s, by rewrite [-teq, -seq, hom_on_mul f aG (SsubG sS)],
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show x ∈ f a * f ' S, by rewrite this; apply mul_mem_lcoset _ (mem_image_of_mem _ sS))
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(assume fafs : x ∈ f a * f ' S,
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obtain t [s (sS : s ∈ S) (seq : f s = t)] (teq : f a * t = x), from fafs,
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have x = f (a * s), by rewrite [-teq, -seq, hom_on_mul f aG (SsubG sS)],
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show x ∈ f ' (a * S), by rewrite this; exact mem_image_of_mem _ (mul_mem_lcoset _ sS)))
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proposition image_rcoset_of_is_hom_on {B : Type} [has_mul B] {f : A → B} {S : set A} {a : A}
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{G : set A} (SsubG : S ⊆ G) (aG : a ∈ G) [is_hom_on f G] :
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f ' (S * a) = f ' S * f a :=
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ext (take x, iff.intro
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(assume fas : x ∈ f ' (S * a),
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obtain t [s (sS : s ∈ S) (seq : s * a = t)] (teq : f t = x), from fas,
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have x = f s * f a, by rewrite [-teq, -seq, hom_on_mul f (SsubG sS) aG],
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show x ∈ f ' S * f a, by rewrite this; exact mul_mem_rcoset (mem_image_of_mem _ sS) _)
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(assume fafs : x ∈ f ' S * f a,
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obtain t [s (sS : s ∈ S) (seq : f s = t)] (teq : t * f a = x), from fafs,
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have x = f (s * a), by rewrite [-teq, -seq, hom_on_mul f (SsubG sS) aG],
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show x ∈ f ' (S * a), by rewrite this; exact mem_image_of_mem _ (mul_mem_rcoset sS _)))
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proposition image_lcoset_of_is_hom {B : Type} [has_mul B] (f : A → B) (a : A) (S : set A)
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[is_hom f] :
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f ' (a * S) = f a * f ' S :=
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have is_hom_on f univ, from is_hom_on_of_is_hom f univ,
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image_lcoset_of_is_hom_on (subset_univ S) !mem_univ
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proposition image_rcoset_of_is_hom {B : Type} [has_mul B] (f : A → B) (S : set A) (a : A)
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[is_hom f] :
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f ' (S * a) = f ' S * f a :=
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have is_hom_on f univ, from is_hom_on_of_is_hom f univ,
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image_rcoset_of_is_hom_on (subset_univ S) !mem_univ
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end has_mulA
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section semigroupA
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variable [semigroup A]
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proposition rcoset_rcoset (S : set A) (a b : A) : S * a * b = S * (a * b) :=
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have H : (mul^~ b) ∘ (mul^~ a) = mul^~ (a * b), from funext (take x, !mul.assoc),
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calc
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S * a * b = ((mul^~ b) ∘ (mul^~ a)) 'S : image_comp
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... = S * (a * b) : by rewrite [↑rcoset, H]
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proposition lcoset_lcoset (S : set A) (a b : A) : a * (b * S) = (a * b) * S :=
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have H : (mul a) ∘ (mul b) = mul (a * b), from funext (take x, !mul.assoc⁻¹),
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calc
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a * (b * S) = ((mul a) ∘ (mul b)) 'S : image_comp
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... = (a * b) * S : by rewrite [↑lcoset, H]
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proposition lcoset_rcoset [semigroup A] (S : set A) (a b : A) : a * S * b = a * (S * b) :=
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have H : (mul^~ b) ∘ (mul a) = (mul a) ∘ (mul^~ b), from funext (take x, !mul.assoc),
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calc
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a * S * b = ((mul^~ b) ∘ (mul a)) 'S : image_comp
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... = ((mul a) ∘ (mul^~ b)) 'S : H
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... = a * (S * b) : image_comp
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end semigroupA
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section monoidA
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variable [monoid A]
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proposition one_lcoset (S : set A) : 1 * S = S :=
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ext (take x, iff.intro
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(suppose x ∈ 1 * S,
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obtain s (sS : s ∈ S) (eqx : 1 * s = x), from this,
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show x ∈ S, by rewrite [-eqx, one_mul]; apply sS)
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(suppose x ∈ S,
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have 1 * x ∈ 1 * S, from mem_image_of_mem (mul 1) this,
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show x ∈ 1 * S, by rewrite one_mul at this; apply this))
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proposition rcoset_one (S : set A) : S * 1 = S :=
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ext (take x, iff.intro
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(suppose x ∈ S * 1,
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obtain s (sS : s ∈ S) (eqx : s * 1 = x), from this,
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show x ∈ S, by rewrite [-eqx, mul_one]; apply sS)
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(suppose x ∈ S,
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have x * 1 ∈ S * 1, from mem_image_of_mem (mul^~ 1) this,
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show x ∈ S * 1, by rewrite mul_one at this; apply this))
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end monoidA
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section groupA
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variable [group A]
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proposition lcoset_inv_lcoset (a : A) (S : set A) : a * (a⁻¹ * S) = S :=
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by rewrite [lcoset_lcoset, mul.right_inv, one_lcoset]
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proposition inv_lcoset_lcoset (a : A) (S : set A) : a⁻¹ * (a * S) = S :=
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by rewrite [lcoset_lcoset, mul.left_inv, one_lcoset]
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proposition rcoset_inv_rcoset (S : set A) (a : A) : (S * a⁻¹) * a = S :=
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by rewrite [rcoset_rcoset, mul.left_inv, rcoset_one]
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proposition rcoset_rcoset_inv (S : set A) (a : A) : (S * a) * a⁻¹ = S :=
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by rewrite [rcoset_rcoset, mul.right_inv, rcoset_one]
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proposition eq_of_lcoset_eq_lcoset {a : A} {S T : set A} (H : a * S = a * T) : S = T :=
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by rewrite [-inv_lcoset_lcoset a S, -inv_lcoset_lcoset a T, H]
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proposition eq_of_rcoset_eq_rcoset {a : A} {S T : set A} (H : S * a = T * a) : S = T :=
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by rewrite [-rcoset_rcoset_inv S a, -rcoset_rcoset_inv T a, H]
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proposition mem_of_mul_mem_lcoset {a b : A} {S : set A} (abaS : a * b ∈ a * S) : b ∈ S :=
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have a⁻¹ * (a * b) ∈ a⁻¹ * (a * S), from mul_mem_lcoset _ abaS,
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by rewrite [inv_mul_cancel_left at this, inv_lcoset_lcoset at this]; apply this
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proposition mul_mem_lcoset_iff (a b : A) (S : set A) : a * b ∈ a * S ↔ b ∈ S :=
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iff.intro !mem_of_mul_mem_lcoset !mul_mem_lcoset
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proposition mem_of_mul_mem_rcoset {a b : A} {S : set A} (abSb : a * b ∈ S * b) : a ∈ S :=
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have (a * b) * b⁻¹ ∈ (S * b) * b⁻¹, from mul_mem_rcoset abSb _,
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by rewrite [mul_inv_cancel_right at this, rcoset_rcoset_inv at this]; apply this
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proposition mul_mem_rcoset_iff (a b : A) (S : set A) : a * b ∈ S * b ↔ a ∈ S :=
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iff.intro !mem_of_mul_mem_rcoset (λ H, mul_mem_rcoset H _)
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proposition inv_mul_mem_of_mem_lcoset {a b : A} {S : set A} (abS : a ∈ b * S) : b⁻¹ * a ∈ S :=
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have b⁻¹ * a ∈ b⁻¹ * (b * S), from mul_mem_lcoset b⁻¹ abS,
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by rewrite inv_lcoset_lcoset at this; apply this
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proposition mem_lcoset_of_inv_mul_mem {a b : A} {S : set A} (H : b⁻¹ * a ∈ S) : a ∈ b * S :=
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have b * (b⁻¹ * a) ∈ b * S, from mul_mem_lcoset b H,
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by rewrite mul_inv_cancel_left at this; apply this
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proposition mem_lcoset_iff (a b : A) (S : set A) : a ∈ b * S ↔ b⁻¹ * a ∈ S :=
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iff.intro inv_mul_mem_of_mem_lcoset mem_lcoset_of_inv_mul_mem
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proposition mul_inv_mem_of_mem_rcoset {a b : A} {S : set A} (aSb : a ∈ S * b) : a * b⁻¹ ∈ S :=
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have a * b⁻¹ ∈ (S * b) * b⁻¹, from mul_mem_rcoset aSb b⁻¹,
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by rewrite rcoset_rcoset_inv at this; apply this
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proposition mem_rcoset_of_mul_inv_mem {a b : A} {S : set A} (H : a * b⁻¹ ∈ S) : a ∈ S * b :=
|
||||
have a * b⁻¹ * b ∈ S * b, from mul_mem_rcoset H b,
|
||||
by rewrite inv_mul_cancel_right at this; apply this
|
||||
|
||||
proposition mem_rcoset_iff (a b : A) (S : set A) : a ∈ S * b ↔ a * b⁻¹ ∈ S :=
|
||||
iff.intro mul_inv_mem_of_mem_rcoset mem_rcoset_of_mul_inv_mem
|
||||
|
||||
proposition lcoset_eq_iff_eq_inv_lcoset (a : A) (S T : set A) : (a * S = T) ↔ (S = a⁻¹ * T) :=
|
||||
iff.intro (assume H, by rewrite [-H, inv_lcoset_lcoset])
|
||||
(assume H, by rewrite [H, lcoset_inv_lcoset])
|
||||
|
||||
proposition rcoset_eq_iff_eq_rcoset_inv (a : A) (S T : set A) : (S * a = T) ↔ (S = T * a⁻¹) :=
|
||||
iff.intro (assume H, by rewrite [-H, rcoset_rcoset_inv])
|
||||
(assume H, by rewrite [H, rcoset_inv_rcoset])
|
||||
|
||||
proposition lcoset_inter (a : A) (S T : set A) [is_subgroup S] [is_subgroup T] :
|
||||
a * (S ∩ T) = (a * S) ∩ (a * T) :=
|
||||
eq_of_subset_of_subset
|
||||
(image_inter_subset _ S T)
|
||||
(take b, suppose b ∈ (a * S) ∩ (a * T),
|
||||
obtain [s [smem (seq : a * s = b)]] [t [tmem (teq : a * t = b)]], from this,
|
||||
have s = t, from eq_of_mul_eq_mul_left' (eq.trans seq (eq.symm teq)),
|
||||
show b ∈ a * (S ∩ T),
|
||||
begin
|
||||
rewrite -seq,
|
||||
apply mul_mem_lcoset,
|
||||
apply and.intro smem,
|
||||
rewrite this, apply tmem
|
||||
end)
|
||||
|
||||
proposition inter_rcoset (a : A) (S T : set A) [is_subgroup S] [is_subgroup T] :
|
||||
(S ∩ T) * a = (S * a) ∩ (T * a) :=
|
||||
eq_of_subset_of_subset
|
||||
(image_inter_subset _ S T)
|
||||
(take b, suppose b ∈ (S * a) ∩ (T * a),
|
||||
obtain [s [smem (seq : s * a = b)]] [t [tmem (teq : t * a = b)]], from this,
|
||||
have s = t, from eq_of_mul_eq_mul_right' (eq.trans seq (eq.symm teq)),
|
||||
show b ∈ (S ∩ T) * a,
|
||||
begin
|
||||
rewrite -seq,
|
||||
apply mul_mem_rcoset,
|
||||
apply and.intro smem,
|
||||
rewrite this, apply tmem
|
||||
end)
|
||||
end groupA
|
||||
|
||||
section subgroupG
|
||||
variables [group A] {G : set A} [is_subgroup G]
|
||||
|
||||
proposition lcoset_eq_self_of_mem {a : A} (aG : a ∈ G) : a * G = G :=
|
||||
ext (take x, iff.intro
|
||||
(assume xaG, obtain g [gG xeq], from xaG,
|
||||
show x ∈ G, by rewrite -xeq; exact (mul_mem aG gG))
|
||||
(assume xG, show x ∈ a * G, from mem_image
|
||||
(show a⁻¹ * x ∈ G, from (mul_mem (inv_mem aG) xG)) !mul_inv_cancel_left))
|
||||
|
||||
proposition rcoset_eq_self_of_mem {a : A} (aG : a ∈ G) : G * a = G :=
|
||||
ext (take x, iff.intro
|
||||
(assume xGa, obtain g [gG xeq], from xGa,
|
||||
show x ∈ G, by rewrite -xeq; exact (mul_mem gG aG))
|
||||
(assume xG, show x ∈ G * a, from mem_image
|
||||
(show x * a⁻¹ ∈ G, from (mul_mem xG (inv_mem aG))) !inv_mul_cancel_right))
|
||||
|
||||
proposition mem_lcoset_self (a : A) : a ∈ a * G :=
|
||||
by rewrite [-mul_one a at {1}]; exact mul_mem_lcoset a one_mem
|
||||
|
||||
proposition mem_rcoset_self (a : A) : a ∈ G * a :=
|
||||
by rewrite [-one_mul a at {1}]; exact mul_mem_rcoset one_mem a
|
||||
|
||||
proposition mem_of_lcoset_eq_self {a : A} (H : a * G = G) : a ∈ G :=
|
||||
by rewrite [-H]; exact mem_lcoset_self a
|
||||
|
||||
proposition mem_of_rcoset_eq_self {a : A} (H : G * a = G) : a ∈ G :=
|
||||
by rewrite [-H]; exact mem_rcoset_self a
|
||||
|
||||
variable (G)
|
||||
|
||||
proposition lcoset_eq_self_iff (a : A) : a * G = G ↔ a ∈ G :=
|
||||
iff.intro mem_of_lcoset_eq_self lcoset_eq_self_of_mem
|
||||
|
||||
proposition rcoset_eq_self_iff (a : A) : G * a = G ↔ a ∈ G :=
|
||||
iff.intro mem_of_rcoset_eq_self rcoset_eq_self_of_mem
|
||||
|
||||
variable {G}
|
||||
|
||||
proposition lcoset_eq_lcoset {a b : A} (H : b⁻¹ * a ∈ G) : a * G = b * G :=
|
||||
have b⁻¹ * (a * G) = b⁻¹ * (b * G),
|
||||
by rewrite [inv_lcoset_lcoset, lcoset_lcoset, lcoset_eq_self_of_mem H],
|
||||
eq_of_lcoset_eq_lcoset this
|
||||
|
||||
proposition inv_mul_mem_of_lcoset_eq_lcoset {a b : A} (H : a * G = b * G) : b⁻¹ * a ∈ G :=
|
||||
mem_of_lcoset_eq_self (by rewrite [-lcoset_lcoset, H, inv_lcoset_lcoset])
|
||||
|
||||
proposition lcoset_eq_lcoset_iff (a b : A) : a * G = b * G ↔ b⁻¹ * a ∈ G :=
|
||||
iff.intro inv_mul_mem_of_lcoset_eq_lcoset lcoset_eq_lcoset
|
||||
|
||||
proposition rcoset_eq_rcoset {a b : A} (H : a * b⁻¹ ∈ G) : G * a = G * b :=
|
||||
have G * a * b⁻¹ = G * b * b⁻¹,
|
||||
by rewrite [rcoset_rcoset_inv, rcoset_rcoset, rcoset_eq_self_of_mem H],
|
||||
eq_of_rcoset_eq_rcoset this
|
||||
|
||||
proposition mul_inv_mem_of_rcoset_eq_rcoset {a b : A} (H : G * a = G * b) : a * b⁻¹ ∈ G :=
|
||||
mem_of_rcoset_eq_self (by rewrite [-rcoset_rcoset, H, rcoset_rcoset_inv])
|
||||
|
||||
proposition rcoset_eq_rcoset_iff (a b : A) : G * a = G * b ↔ a * b⁻¹ ∈ G :=
|
||||
iff.intro mul_inv_mem_of_rcoset_eq_rcoset rcoset_eq_rcoset
|
||||
end subgroupG
|
||||
|
||||
|
||||
/- normal cosets and the normalizer -/
|
||||
|
||||
section has_mulA
|
||||
variable [has_mul A]
|
||||
|
||||
abbreviation normalizes [reducible] (a : A) (S : set A) : Prop := a * S = S * a
|
||||
|
||||
definition is_normal [class] (S : set A) : Prop := ∀ a, normalizes a S
|
||||
|
||||
definition normalizer (S : set A) : set A := { a : A | normalizes a S }
|
||||
|
||||
definition is_normal_in [class] (S T : set A) : Prop := T ⊆ normalizer S
|
||||
|
||||
abbreviation normalizer_in [reducible] (S T : set A) : set A := T ∩ normalizer S
|
||||
|
||||
proposition lcoset_eq_rcoset (a : A) (S : set A) [H : is_normal S] : a * S = S * a := H a
|
||||
|
||||
proposition subset_normalizer (S T : set A) [H : is_normal_in T S] : S ⊆ normalizer T := H
|
||||
|
||||
proposition lcoset_eq_rcoset_of_mem {a : A} (S : set A) {T : set A} [H : is_normal_in S T]
|
||||
(amemT : a ∈ T) :
|
||||
a * S = S * a := H amemT
|
||||
|
||||
proposition is_normal_in_of_is_normal (S T : set A) [H : is_normal S] : is_normal_in S T :=
|
||||
forallb_of_forall T H
|
||||
|
||||
proposition is_normal_of_is_normal_in_univ {S : set A} (H : is_normal_in S univ) :
|
||||
is_normal S :=
|
||||
forall_of_forallb_univ H
|
||||
|
||||
proposition is_normal_in_univ_iff_is_normal (S : set A) : is_normal_in S univ ↔ is_normal S :=
|
||||
forallb_univ_iff_forall _
|
||||
|
||||
proposition is_normal_in_of_subset {S T U : set A} (H : T ⊆ U) (H' : is_normal_in S U) :
|
||||
is_normal_in S T :=
|
||||
forallb_of_subset H H'
|
||||
|
||||
proposition normalizes_of_mem_normalizer {a : A} {S : set A} (H : a ∈ normalizer S) :
|
||||
normalizes a S := H
|
||||
|
||||
proposition mem_normalizer_iff_normalizes (a : A) (S : set A) :
|
||||
a ∈ normalizer S ↔ normalizes a S := iff.refl _
|
||||
|
||||
proposition is_normal_in_normalizer [instance] (S : set A) : is_normal_in S (normalizer S) :=
|
||||
subset.refl (normalizer S)
|
||||
end has_mulA
|
||||
|
||||
section groupA
|
||||
variable [group A]
|
||||
|
||||
proposition is_normal_in_of_forall_subset {S G : set A} [is_subgroup G]
|
||||
(H : ∀₀ x ∈ G, x * S ⊆ S * x) :
|
||||
is_normal_in S G :=
|
||||
take x, assume xG,
|
||||
show x * S = S * x, from eq_of_subset_of_subset (H xG)
|
||||
(have x * (x⁻¹ * S) * x ⊆ x * (S * x⁻¹) * x,
|
||||
from rcoset_subset_rcoset (lcoset_subset_lcoset x (H (inv_mem xG))) x,
|
||||
show S * x ⊆ x * S,
|
||||
begin
|
||||
rewrite [lcoset_inv_lcoset at this, lcoset_rcoset at this, rcoset_inv_rcoset at this],
|
||||
exact this
|
||||
end)
|
||||
|
||||
proposition is_normal_of_forall_subset {S : set A} (H : ∀ x, x * S ⊆ S * x) : is_normal S :=
|
||||
begin
|
||||
rewrite [-is_normal_in_univ_iff_is_normal],
|
||||
apply is_normal_in_of_forall_subset,
|
||||
intro x xuniv, exact H x
|
||||
end
|
||||
|
||||
proposition subset_normalizer_self (G : set A) [is_subgroup G] : G ⊆ normalizer G :=
|
||||
take a, assume aG, show a * G = G * a,
|
||||
by rewrite [lcoset_eq_self_of_mem aG, rcoset_eq_self_of_mem aG]
|
||||
end groupA
|
||||
|
||||
section normalG
|
||||
variables [group A] (G : set A) [is_normal G]
|
||||
|
||||
proposition lcoset_equiv_mul {a₁ a₂ b₁ b₂ : A}
|
||||
(H₁ : lcoset_equiv G a₁ a₂) (H₂ : lcoset_equiv G b₁ b₂) : lcoset_equiv G (a₁ * b₁) (a₂ * b₂) :=
|
||||
begin
|
||||
unfold lcoset_equiv at *,
|
||||
rewrite [-lcoset_lcoset, H₂, lcoset_eq_rcoset, -lcoset_rcoset, H₁, lcoset_rcoset,
|
||||
-lcoset_eq_rcoset, lcoset_lcoset]
|
||||
end
|
||||
|
||||
proposition lcoset_equiv_inv {a₁ a₂ : A} (H : lcoset_equiv G a₁ a₂) : lcoset_equiv G a₁⁻¹ a₂⁻¹ :=
|
||||
begin
|
||||
unfold lcoset_equiv at *,
|
||||
have a₁⁻¹ * G = a₂⁻¹ * (a₂ * G) * a₁⁻¹, by rewrite [inv_lcoset_lcoset, lcoset_eq_rcoset],
|
||||
rewrite [this, -H, lcoset_rcoset, lcoset_eq_rcoset, rcoset_rcoset_inv]
|
||||
end
|
||||
end normalG
|
||||
|
||||
|
||||
/- the normalizer is a subgroup -/
|
||||
|
||||
section semigroupA
|
||||
variable [semigroup A]
|
||||
|
||||
proposition mul_mem_normalizer {S : set A} {a b : A}
|
||||
(Ha : a ∈ normalizer S) (Hb : b ∈ normalizer S) : a * b ∈ normalizer S :=
|
||||
show a * b * S = S * (a * b),
|
||||
by rewrite [-lcoset_lcoset, normalizes_of_mem_normalizer Hb, -lcoset_rcoset,
|
||||
normalizes_of_mem_normalizer Ha, rcoset_rcoset]
|
||||
end semigroupA
|
||||
|
||||
section monoidA
|
||||
variable [monoid A]
|
||||
|
||||
proposition one_mem_normalizer (S : set A) : 1 ∈ normalizer S :=
|
||||
by rewrite [↑normalizer, mem_set_of_iff, one_lcoset, rcoset_one]
|
||||
end monoidA
|
||||
|
||||
section groupA
|
||||
variable [group A]
|
||||
|
||||
proposition inv_mem_normalizer {S : set A} {a : A} (H : a ∈ normalizer S) : a⁻¹ ∈ normalizer S :=
|
||||
have a⁻¹ * S = S * a⁻¹,
|
||||
begin
|
||||
apply iff.mp (rcoset_eq_iff_eq_rcoset_inv _ _ _),
|
||||
rewrite [lcoset_rcoset, -normalizes_of_mem_normalizer H, inv_lcoset_lcoset]
|
||||
end,
|
||||
by rewrite [↑normalizer, mem_set_of_iff, this]
|
||||
|
||||
proposition is_subgroup_normalizer [instance] (S : set A) : is_subgroup (normalizer S) :=
|
||||
⦃ is_subgroup,
|
||||
one_mem := one_mem_normalizer S,
|
||||
mul_mem := λ a Ha b Hb, mul_mem_normalizer Ha Hb,
|
||||
inv_mem := λ a H, inv_mem_normalizer H⦄
|
||||
end groupA
|
||||
|
||||
section subgroupG
|
||||
variables [group A] {G : set A} [is_subgroup G]
|
||||
|
||||
proposition normalizes_image_of_is_hom_on [group B] {a : A} (aG : a ∈ G) {S : set A}
|
||||
(SsubG : S ⊆ G) (H : normalizes a S) (f : A → B) [is_hom_on f G] :
|
||||
normalizes (f a) (f ' S) :=
|
||||
by rewrite [-image_lcoset_of_is_hom_on SsubG aG, -image_rcoset_of_is_hom_on SsubG aG,
|
||||
↑normalizes at H, H]
|
||||
|
||||
proposition is_normal_in_image_image [group B] {S T : set A} (SsubT : S ⊆ T)
|
||||
[H : is_normal_in S T] (f : A → B) [is_subgroup T] [is_hom_on f T] :
|
||||
is_normal_in (f ' S) (f ' T) :=
|
||||
take a, assume afT,
|
||||
obtain b [bT (beq : f b = a)], from afT,
|
||||
show normalizes a (f ' S),
|
||||
begin rewrite -beq, apply (normalizes_image_of_is_hom_on bT SsubT (H bT)) end
|
||||
|
||||
proposition normalizes_image_of_is_hom [group B] {a : A} {S : set A}
|
||||
(H : normalizes a S) (f : A → B) [is_hom f] :
|
||||
normalizes (f a) (f ' S) :=
|
||||
by rewrite [-image_lcoset_of_is_hom f a S, -image_rcoset_of_is_hom f S a,
|
||||
↑normalizes at H, H]
|
||||
|
||||
proposition is_normal_in_image_image_univ [group B] {S : set A}
|
||||
[H : is_normal S] (f : A → B) [is_hom f] :
|
||||
is_normal_in (f ' S) (f ' univ) :=
|
||||
take a, assume afT,
|
||||
obtain b [buniv (beq : f b = a)], from afT,
|
||||
show normalizes a (f ' S),
|
||||
begin rewrite -beq, apply (normalizes_image_of_is_hom (H b) f) end
|
||||
end subgroupG
|
||||
|
||||
|
||||
/- conjugation -/
|
||||
|
||||
definition conj [reducible] [group A] (a b : A) : A := b⁻¹ * a * b
|
||||
|
||||
definition set_conj [reducible] [group A] (S : set A)(a : A) : set A := a⁻¹ * S * a
|
||||
-- conj^~ a ' S
|
||||
|
||||
namespace conj_notation
|
||||
infix `^` := conj
|
||||
infix `^` := set_conj
|
||||
end conj_notation
|
||||
|
||||
open conj_notation
|
||||
|
||||
section groupA
|
||||
variables [group A]
|
||||
|
||||
proposition set_conj_eq_image_conj (S : set A) (a : A) : S^a = conj^~ a 'S :=
|
||||
eq.symm !image_comp
|
||||
|
||||
proposition set_conj_eq_self_of_normalizes {S : set A} {a : A} (H : normalizes a S) : S^a = S :=
|
||||
by rewrite [lcoset_rcoset, ↑normalizes at H, -H, inv_lcoset_lcoset]
|
||||
|
||||
proposition normalizes_of_set_conj_eq_self {S : set A} {a : A} (H : S^a = S) : normalizes a S :=
|
||||
by rewrite [-H at {1}, ↑set_conj, lcoset_rcoset, lcoset_inv_lcoset]
|
||||
|
||||
proposition set_conj_eq_self_iff_normalizes (S : set A) (a : A) : S^a = S ↔ normalizes a S :=
|
||||
iff.intro normalizes_of_set_conj_eq_self set_conj_eq_self_of_normalizes
|
||||
|
||||
proposition set_conj_eq_self_of_mem_normalizer {S : set A} {a : A} (H : a ∈ normalizer S) :
|
||||
S^a = S := set_conj_eq_self_of_normalizes H
|
||||
|
||||
proposition mem_normalizer_of_set_conj_eq_self {S : set A} {a : A} (H : S^a = S) :
|
||||
a ∈ normalizer S := normalizes_of_set_conj_eq_self H
|
||||
|
||||
proposition set_conj_eq_self_iff_mem_normalizer (S : set A) (a : A) :
|
||||
S^a = S ↔ a ∈ normalizer S :=
|
||||
iff.intro mem_normalizer_of_set_conj_eq_self set_conj_eq_self_of_mem_normalizer
|
||||
|
||||
proposition conj_one (a : A) : a ^ (1 : A) = a :=
|
||||
by rewrite [↑conj, one_inv, one_mul, mul_one]
|
||||
|
||||
proposition conj_conj (a b c : A) : (a^b)^c = a^(b * c) :=
|
||||
by rewrite [↑conj, mul_inv, *mul.assoc]
|
||||
|
||||
proposition conj_inv (a b : A) : (a^b)⁻¹ = (a⁻¹)^b :=
|
||||
by rewrite[mul_inv, mul_inv, inv_inv, mul.assoc]
|
||||
|
||||
proposition mul_conj (a b c : A) : (a * b)^c = a^c * b^c :=
|
||||
by rewrite[↑conj, *mul.assoc, mul_inv_cancel_left]
|
||||
end groupA
|
||||
|
||||
|
||||
/- the kernel -/
|
||||
|
||||
definition ker [has_one B] (f : A → B) : set A := { x | f x = 1 }
|
||||
|
||||
section hasoneB
|
||||
variable [has_one B]
|
||||
|
||||
proposition eq_one_of_mem_ker {f : A → B} {a : A} (H : a ∈ ker f) : f a = 1 := H
|
||||
|
||||
proposition mem_ker_iff (f : A → B) (a : A) : a ∈ ker f ↔ f a = 1 := iff.rfl
|
||||
|
||||
proposition ker_eq_preimage_one (f : A → B) : ker f = f '- '{1} :=
|
||||
ext (take x, by rewrite [mem_ker_iff, -mem_preimage_iff, mem_singleton_iff])
|
||||
|
||||
definition ker_in (f : A → B) (S : set A) : set A := ker f ∩ S
|
||||
|
||||
proposition ker_in_univ (f : A → B) : ker_in f univ = ker f :=
|
||||
!inter_univ
|
||||
end hasoneB
|
||||
|
||||
section groupAB
|
||||
variables [group A] [group B]
|
||||
variable {f : A → B}
|
||||
|
||||
proposition eq_of_mul_inv_mem_ker [is_hom f] {a₁ a₂ : A} (H : a₁ * a₂⁻¹ ∈ ker f) :
|
||||
f a₁ = f a₂ :=
|
||||
eq_of_mul_inv_eq_one (by rewrite [-hom_inv f, -hom_mul f]; exact H)
|
||||
|
||||
proposition mul_inv_mem_ker_of_eq [is_hom f] {a₁ a₂ : A} (H : f a₁ = f a₂) :
|
||||
a₁ * a₂⁻¹ ∈ ker f :=
|
||||
show f (a₁ * a₂⁻¹) = 1, by rewrite [hom_mul f, hom_inv f, H, mul.right_inv]
|
||||
|
||||
proposition eq_iff_mul_inv_mem_ker [is_hom f] (a₁ a₂ : A) : f a₁ = f a₂ ↔ a₁ * a₂⁻¹ ∈ ker f :=
|
||||
iff.intro mul_inv_mem_ker_of_eq eq_of_mul_inv_mem_ker
|
||||
|
||||
proposition eq_of_mul_inv_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : a₁ * a₂⁻¹ ∈ ker_in f G) :
|
||||
f a₁ = f a₂ :=
|
||||
eq_of_mul_inv_eq_one (by rewrite [-hom_on_inv f a₂G, -hom_on_mul f a₁G (inv_mem a₂G)];
|
||||
exact and.left H)
|
||||
|
||||
proposition mul_inv_mem_ker_in_of_eq {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : f a₁ = f a₂) :
|
||||
a₁ * a₂⁻¹ ∈ ker_in f G :=
|
||||
and.intro
|
||||
(show f (a₁ * a₂⁻¹) = 1,
|
||||
by rewrite [hom_on_mul f a₁G (inv_mem a₂G), hom_on_inv f a₂G, H, mul.right_inv])
|
||||
(mul_mem a₁G (inv_mem a₂G))
|
||||
|
||||
proposition eq_iff_mul_inv_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) :
|
||||
f a₁ = f a₂ ↔ a₁ * a₂⁻¹ ∈ ker_in f G :=
|
||||
iff.intro (mul_inv_mem_ker_in_of_eq a₁G a₂G) (eq_of_mul_inv_mem_ker_in a₁G a₂G)
|
||||
|
||||
-- Ouch! These versions are not equivalent to the ones before.
|
||||
|
||||
proposition eq_of_inv_mul_mem_ker [is_hom f] {a₁ a₂ : A} (H : a₁⁻¹ * a₂ ∈ ker f) :
|
||||
f a₁ = f a₂ :=
|
||||
eq.symm (eq_of_inv_mul_eq_one (by rewrite [-hom_inv f, -hom_mul f]; exact H))
|
||||
|
||||
proposition inv_mul_mem_ker_of_eq [is_hom f] {a₁ a₂ : A} (H : f a₁ = f a₂) :
|
||||
a₁⁻¹ * a₂ ∈ ker f :=
|
||||
show f (a₁⁻¹ * a₂) = 1, by rewrite [hom_mul f, hom_inv f, H, mul.left_inv]
|
||||
|
||||
proposition eq_iff_inv_mul_mem_ker [is_hom f] (a₁ a₂ : A) : f a₁ = f a₂ ↔ a₁⁻¹ * a₂ ∈ ker f :=
|
||||
iff.intro inv_mul_mem_ker_of_eq eq_of_inv_mul_mem_ker
|
||||
|
||||
proposition eq_of_inv_mul_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : a₁⁻¹ * a₂ ∈ ker_in f G) :
|
||||
f a₁ = f a₂ :=
|
||||
eq.symm (eq_of_inv_mul_eq_one (by rewrite [-hom_on_inv f a₁G, -hom_on_mul f (inv_mem a₁G) a₂G];
|
||||
exact and.left H))
|
||||
|
||||
proposition inv_mul_mem_ker_in_of_eq {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) (H : f a₁ = f a₂) :
|
||||
a₁⁻¹ * a₂ ∈ ker_in f G :=
|
||||
and.intro
|
||||
(show f (a₁⁻¹ * a₂) = 1,
|
||||
by rewrite [hom_on_mul f (inv_mem a₁G) a₂G, hom_on_inv f a₁G, H, mul.left_inv])
|
||||
(mul_mem (inv_mem a₁G) a₂G)
|
||||
|
||||
proposition eq_iff_inv_mul_mem_ker_in {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
{a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G) :
|
||||
f a₁ = f a₂ ↔ a₁⁻¹ * a₂ ∈ ker_in f G :=
|
||||
iff.intro (inv_mul_mem_ker_in_of_eq a₁G a₂G) (eq_of_inv_mul_mem_ker_in a₁G a₂G)
|
||||
|
||||
proposition eq_one_of_eq_one_of_injective [is_hom f] (H : injective f) {x : A}
|
||||
(H' : f x = 1) :
|
||||
x = 1 :=
|
||||
H (by rewrite [H', hom_one f])
|
||||
|
||||
proposition eq_one_iff_eq_one_of_injective [is_hom f] (H : injective f) (x : A) :
|
||||
f x = 1 ↔ x = 1 :=
|
||||
iff.intro (eq_one_of_eq_one_of_injective H) (λ H', by rewrite [H', hom_one f])
|
||||
|
||||
proposition injective_of_forall_eq_one [is_hom f] (H : ∀ x, f x = 1 → x = 1) : injective f :=
|
||||
take a₁ a₂, assume Heq,
|
||||
have f (a₁ * a₂⁻¹) = 1, by rewrite [hom_mul f, hom_inv f, Heq, mul.right_inv],
|
||||
eq_of_mul_inv_eq_one (H _ this)
|
||||
|
||||
proposition injective_of_ker_eq_singleton_one [is_hom f] (H : ker f = '{1}) : injective f :=
|
||||
injective_of_forall_eq_one
|
||||
(take x, suppose x ∈ ker f, by rewrite [H at this]; exact eq_of_mem_singleton this)
|
||||
|
||||
proposition ker_eq_singleton_one_of_injective [is_hom f] (H : injective f) : ker f = '{1} :=
|
||||
ext (take x, by rewrite [mem_ker_iff, mem_singleton_iff, eq_one_iff_eq_one_of_injective H])
|
||||
|
||||
variable (f)
|
||||
proposition injective_iff_ker_eq_singleton_one [is_hom f] : injective f ↔ ker f = '{1} :=
|
||||
iff.intro ker_eq_singleton_one_of_injective injective_of_ker_eq_singleton_one
|
||||
variable {f}
|
||||
|
||||
proposition eq_one_of_eq_one_of_inj_on {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
(H : inj_on f G) {x : A} (xG : x ∈ G) (H' : f x = 1) :
|
||||
x = 1 :=
|
||||
H xG one_mem (by rewrite [H', hom_on_one f G])
|
||||
|
||||
proposition eq_one_iff_eq_one_of_inj_on {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
(H : inj_on f G) {x : A} (xG : x ∈ G) [is_hom_on f G] :
|
||||
f x = 1 ↔ x = 1 :=
|
||||
iff.intro (eq_one_of_eq_one_of_inj_on H xG) (λ H', by rewrite [H', hom_on_one f G])
|
||||
|
||||
proposition inj_on_of_forall_eq_one {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
(H : ∀₀ x ∈ G, f x = 1 → x = 1) : inj_on f G :=
|
||||
take a₁ a₂, assume a₁G a₂G Heq,
|
||||
have f (a₁ * a₂⁻¹) = 1,
|
||||
by rewrite [hom_on_mul f a₁G (inv_mem a₂G), hom_on_inv f a₂G, Heq, mul.right_inv],
|
||||
eq_of_mul_inv_eq_one (H (mul_mem a₁G (inv_mem a₂G)) this)
|
||||
|
||||
proposition inj_on_of_ker_in_eq_singleton_one {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
(H : ker_in f G = '{1}) : inj_on f G :=
|
||||
inj_on_of_forall_eq_one
|
||||
(take x, assume xG fxone,
|
||||
have x ∈ ker_in f G, from and.intro fxone xG,
|
||||
by rewrite [H at this]; exact eq_of_mem_singleton this)
|
||||
|
||||
proposition ker_in_eq_singleton_one_of_inj_on {G : set A} [is_subgroup G] [is_hom_on f G]
|
||||
(H : inj_on f G) : ker_in f G = '{1} :=
|
||||
ext (take x,
|
||||
begin
|
||||
rewrite [↑ker_in, mem_inter_iff, mem_ker_iff, mem_singleton_iff],
|
||||
apply iff.intro,
|
||||
{intro H', cases H' with fxone xG, exact eq_one_of_eq_one_of_inj_on H xG fxone},
|
||||
intro xone, rewrite xone, split, exact hom_on_one f G, exact one_mem
|
||||
end)
|
||||
|
||||
variable (f)
|
||||
proposition inj_on_iff_ker_in_eq_singleton_one (G : set A) [is_subgroup G] [is_hom_on f G] :
|
||||
inj_on f G ↔ ker_in f G = '{1} :=
|
||||
iff.intro ker_in_eq_singleton_one_of_inj_on inj_on_of_ker_in_eq_singleton_one
|
||||
variable {f}
|
||||
|
||||
proposition conj_mem_ker [is_hom f] {a₁ : A} (a₂ : A) (H : a₁ ∈ ker f) : a₁^a₂ ∈ ker f :=
|
||||
show f (a₁^a₂) = 1,
|
||||
by rewrite [↑conj, *(hom_mul f), hom_inv f, eq_one_of_mem_ker H, mul_one, mul.left_inv]
|
||||
|
||||
variable (f)
|
||||
|
||||
proposition is_subgroup_ker_in [instance] (S : set A) [is_subgroup S] [is_hom_on f S] :
|
||||
is_subgroup (ker_in f S) :=
|
||||
⦃ is_subgroup,
|
||||
one_mem := and.intro (hom_on_one f S) one_mem,
|
||||
mul_mem := λ a aker b bker,
|
||||
obtain (fa : f a = 1) (aS : a ∈ S), from aker,
|
||||
obtain (fb : f b = 1) (bS : b ∈ S), from bker,
|
||||
and.intro (show f (a * b) = 1, by rewrite [hom_on_mul f aS bS, fa, fb, one_mul])
|
||||
(mul_mem aS bS),
|
||||
inv_mem := λ a aker,
|
||||
obtain (fa : f a = 1) (aS : a ∈ S), from aker,
|
||||
and.intro (show f (a⁻¹) = 1, by rewrite [hom_on_inv f aS, fa, one_inv])
|
||||
(inv_mem aS)
|
||||
⦄
|
||||
|
||||
proposition is_subgroup_ker [instance] [is_hom f] : is_subgroup (ker f) :=
|
||||
begin
|
||||
rewrite [-ker_in_univ f],
|
||||
have is_hom_on f univ, from is_hom_on_of_is_hom f univ,
|
||||
apply is_subgroup_ker_in f univ
|
||||
end
|
||||
|
||||
proposition is_normal_in_ker_in [instance] (G : set A) [is_subgroup G] [is_hom_on f G] :
|
||||
is_normal_in (ker_in f G) G :=
|
||||
is_normal_in_of_forall_subset
|
||||
(take x, assume xG, take y, assume yker,
|
||||
obtain z [[(fz : f z = 1) zG] (yeq : x * z = y)], from yker,
|
||||
have y = x * z * x⁻¹ * x, by rewrite [yeq, inv_mul_cancel_right],
|
||||
show y ∈ ker_in f G * x,
|
||||
begin
|
||||
rewrite this,
|
||||
apply mul_mem_rcoset,
|
||||
apply and.intro,
|
||||
show f (x * z * x⁻¹) = 1,
|
||||
by rewrite [hom_on_mul f (mul_mem xG zG) (inv_mem xG), hom_on_mul f xG zG, fz,
|
||||
hom_on_inv f xG, mul_one, mul.right_inv],
|
||||
show x * z * x⁻¹ ∈ G, from mul_mem (mul_mem xG zG) (inv_mem xG)
|
||||
end)
|
||||
|
||||
proposition is_normal_ker [instance] [H : is_hom f] : is_normal (ker f) :=
|
||||
begin
|
||||
rewrite [-ker_in_univ, -is_normal_in_univ_iff_is_normal],
|
||||
apply is_normal_in_ker_in,
|
||||
exact is_hom_on_of_is_hom f univ
|
||||
end
|
||||
|
||||
end groupAB
|
||||
|
||||
section subgroupH
|
||||
variables [group A] [group B] {H : set A} [is_subgroup H]
|
||||
variables {f : A → B} [is_hom f]
|
||||
|
||||
proposition subset_ker_of_forall (hyp : ∀ x y, x * H = y * H → f x = f y) : H ⊆ ker f :=
|
||||
take h, assume hH,
|
||||
have h * H = 1 * H, by rewrite [lcoset_eq_self_of_mem hH, one_lcoset],
|
||||
have f h = f 1, from hyp h 1 this,
|
||||
show f h = 1, by rewrite [this, hom_one f]
|
||||
|
||||
proposition eq_of_lcoset_eq_lcoset_of_subset_ker {x y : A} (hyp₀ : x * H = y * H) (hyp₁ : H ⊆ ker f) :
|
||||
f x = f y :=
|
||||
have y⁻¹ * x ∈ H, from inv_mul_mem_of_lcoset_eq_lcoset hyp₀,
|
||||
eq.symm (eq_of_inv_mul_mem_ker (hyp₁ this))
|
||||
|
||||
variables (H f)
|
||||
proposition subset_ker_iff : H ⊆ ker f ↔ ∀ x y, x * H = y * H → f x = f y :=
|
||||
iff.intro (λ h₁ x y h₀, eq_of_lcoset_eq_lcoset_of_subset_ker h₀ h₁) subset_ker_of_forall
|
||||
end subgroupH
|
||||
|
||||
section subgroupGH
|
||||
variables [group A] [group B] {G H : set A} [is_subgroup G] [is_subgroup H]
|
||||
variables {f : A → B} [is_hom_on f G]
|
||||
|
||||
proposition subset_ker_in_of_forall (hyp₀ : ∀₀ x ∈ G, ∀₀ y ∈ G, x * H = y * H → f x = f y)
|
||||
(hyp₁ : H ⊆ G) :
|
||||
H ⊆ ker_in f G :=
|
||||
take h, assume hH,
|
||||
have hG : h ∈ G, from hyp₁ hH,
|
||||
and.intro
|
||||
(have h * H = 1 * H, by rewrite [lcoset_eq_self_of_mem hH, one_lcoset],
|
||||
have f h = f 1, from hyp₀ hG one_mem this,
|
||||
show f h = 1, by rewrite [this, hom_on_one f G])
|
||||
hG
|
||||
|
||||
proposition eq_of_lcoset_eq_lcoset_of_subset_ker_in {x : A} (xG : x ∈ G) {y : A} (yG : y ∈ G)
|
||||
(hyp₀ : x * H = y * H) (hyp₁ : H ⊆ ker_in f G) :
|
||||
f x = f y :=
|
||||
have y⁻¹ * x ∈ H, from inv_mul_mem_of_lcoset_eq_lcoset hyp₀,
|
||||
eq.symm (eq_of_inv_mul_mem_ker_in yG xG (hyp₁ this))
|
||||
|
||||
variables (H f)
|
||||
proposition subset_ker_in_iff :
|
||||
H ⊆ ker_in f G ↔ (H ⊆ G ∧ ∀₀ x ∈ G, ∀₀ y ∈ G, x * H = y * H → f x = f y) :=
|
||||
iff.intro
|
||||
(λ h₁, and.intro
|
||||
(subset.trans h₁ (inter_subset_right _ _))
|
||||
(λ x xG y yG h₀, eq_of_lcoset_eq_lcoset_of_subset_ker_in xG yG h₀ h₁))
|
||||
(λ h, subset_ker_in_of_forall (and.right h) (and.left h))
|
||||
end subgroupGH
|
||||
|
||||
|
||||
/- the centralizer -/
|
||||
|
||||
section has_mulA
|
||||
variable [has_mul A]
|
||||
|
||||
abbreviation centralizes [reducible] (a : A) (S : set A) : Prop := ∀₀ b ∈ S, a * b = b * a
|
||||
|
||||
definition centralizer (S : set A) : set A := { a : A | centralizes a S }
|
||||
|
||||
abbreviation is_centralized_by (S T : set A) : Prop := T ⊆ centralizer S
|
||||
|
||||
abbreviation centralizer_in (S T : set A) : set A := T ∩ centralizer S
|
||||
|
||||
proposition mem_centralizer_iff_centralizes (a : A) (S : set A) :
|
||||
a ∈ centralizer S ↔ centralizes a S := iff.refl _
|
||||
|
||||
proposition normalizes_of_centralizes {a : A} {S : set A} (H : centralizes a S) :
|
||||
normalizes a S :=
|
||||
ext (take b, iff.intro
|
||||
(suppose b ∈ a * S,
|
||||
obtain s [ains (beq : a * s = b)], from this,
|
||||
show b ∈ S * a, by rewrite[-beq, H ains]; apply mem_image_of_mem _ ains)
|
||||
(suppose b ∈ S * a,
|
||||
obtain s [ains (beq : s * a = b)], from this,
|
||||
show b ∈ a * S, by rewrite[-beq, -H ains]; apply mem_image_of_mem _ ains))
|
||||
|
||||
proposition centralizer_subset_normalizer (S : set A) : centralizer S ⊆ normalizer S :=
|
||||
λ a acent, normalizes_of_centralizes acent
|
||||
|
||||
proposition centralizer_subset_centralizer {S T : set A} (ssubt : S ⊆ T) :
|
||||
centralizer T ⊆ centralizer S :=
|
||||
λ x xCentT s sS, xCentT _ (ssubt sS)
|
||||
end has_mulA
|
||||
|
||||
section groupA
|
||||
variable [group A]
|
||||
|
||||
proposition is_subgroup_centralizer [instance] [group A] (S : set A) :
|
||||
is_subgroup (centralizer S) :=
|
||||
⦃ is_subgroup,
|
||||
one_mem := λ b bS, by rewrite [one_mul, mul_one],
|
||||
mul_mem := λ a acent b bcent c cS, by rewrite [mul.assoc, bcent cS, -*mul.assoc, acent cS],
|
||||
inv_mem := λ a acent c cS, eq_mul_inv_of_mul_eq
|
||||
(by rewrite [mul.assoc, -acent cS, inv_mul_cancel_left])⦄
|
||||
end groupA
|
||||
|
||||
|
||||
/- the subgroup generated by a set -/
|
||||
|
||||
section groupA
|
||||
variable [group A]
|
||||
|
||||
inductive subgroup_generated_by (S : set A) : A → Prop :=
|
||||
| generators_mem : ∀ x, x ∈ S → subgroup_generated_by S x
|
||||
| one_mem : subgroup_generated_by S 1
|
||||
| mul_mem : ∀ x y, subgroup_generated_by S x → subgroup_generated_by S y →
|
||||
subgroup_generated_by S (x * y)
|
||||
| inv_mem : ∀ x, subgroup_generated_by S x → subgroup_generated_by S (x⁻¹)
|
||||
|
||||
theorem generators_subset_subgroup_generated_by (S : set A) : S ⊆ subgroup_generated_by S :=
|
||||
subgroup_generated_by.generators_mem
|
||||
|
||||
theorem is_subgroup_subgroup_generated_by [instance] (S : set A) :
|
||||
is_subgroup (subgroup_generated_by S) :=
|
||||
⦃ is_subgroup,
|
||||
one_mem := subgroup_generated_by.one_mem S,
|
||||
mul_mem := λ a amem b bmem, subgroup_generated_by.mul_mem a b amem bmem,
|
||||
inv_mem := λ a amem, subgroup_generated_by.inv_mem a amem ⦄
|
||||
|
||||
theorem subgroup_generated_by_subset {S G : set A} [is_subgroup G] (H : S ⊆ G) :
|
||||
subgroup_generated_by S ⊆ G :=
|
||||
begin
|
||||
intro x xgenS,
|
||||
induction xgenS with a aS a b agen bgen aG bG a agen aG,
|
||||
{exact H aS},
|
||||
{exact one_mem},
|
||||
{exact mul_mem aG bG},
|
||||
exact inv_mem aG
|
||||
end
|
||||
end groupA
|
||||
|
||||
end group_theory
|
||||
6
library/theories/group_theory/group_theory.md
Normal file
6
library/theories/group_theory/group_theory.md
Normal file
|
|
@ -0,0 +1,6 @@
|
|||
theories.group_theory
|
||||
=====================
|
||||
|
||||
* [basic](basic.lean)
|
||||
* [subgroup_to_group](subgroup_to_group.lean)
|
||||
* [quotient](quotient.lean)
|
||||
566
library/theories/group_theory/quotient.lean
Normal file
566
library/theories/group_theory/quotient.lean
Normal file
|
|
@ -0,0 +1,566 @@
|
|||
/-
|
||||
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Andrew Zipperer, Jeremy Avigad
|
||||
|
||||
We provide two versions of the quoptient construction. They use the same names and notation:
|
||||
one lives in the namespace 'quotient_group' and the other lives in the namespace
|
||||
'quotient_group_general'.
|
||||
|
||||
The first takes a group, A, and a normal subgroup, H. We have
|
||||
|
||||
quotient H := the quotient of A by H
|
||||
qproj H a := the projection, with notation a' * G
|
||||
qproj H ' s := the image of s, with notation s / G
|
||||
extend H respf := given f : A → B respecting the equivalence relation, we get a function
|
||||
f : quotient G → B
|
||||
bar f := the above, G = ker f)
|
||||
|
||||
The definition is constructive, using quotient types. We prove all the characteristic properties.
|
||||
|
||||
As in the SSReflect library, we also provide a construction to quotient by an *arbitrary subgroup*.
|
||||
Now we have
|
||||
|
||||
quotient H := the quotient of normalizer H by H
|
||||
qproj H a := still denoted a '* H, the projection when a is in normalizer H,
|
||||
arbitrary otherwise
|
||||
qproj H G := still denoted G / H, the image of the above
|
||||
extend H G respf := given a homomorphism on G with ker_in G f ⊆ H, extends to a
|
||||
homomorphism G / H
|
||||
bar G f := the above, with H = ker_in f G
|
||||
|
||||
This quotient H is defined by composing the first one with the construction which turns
|
||||
normalizer H into a group.
|
||||
-/
|
||||
import .subgroup_to_group theories.move
|
||||
open set function subtype classical quot
|
||||
|
||||
namespace group_theory
|
||||
open coset_notation
|
||||
|
||||
variables {A B C : Type}
|
||||
|
||||
/- the quotient group -/
|
||||
|
||||
namespace quotient_group
|
||||
|
||||
variables [group A] (H : set A) [is_normal H]
|
||||
|
||||
definition lcoset_setoid [instance] : setoid A :=
|
||||
setoid.mk (lcoset_equiv H) (equivalence_lcoset_equiv H)
|
||||
|
||||
definition quotient := quot (lcoset_setoid H)
|
||||
|
||||
private definition qone : quotient H := ⟦ 1 ⟧
|
||||
|
||||
private definition qmul : quotient H → quotient H → quotient H :=
|
||||
quot.lift₂
|
||||
(λ a b, ⟦a * b⟧)
|
||||
(λ a₁ a₂ b₁ b₂ e₁ e₂, quot.sound (lcoset_equiv_mul H e₁ e₂))
|
||||
|
||||
private definition qinv : quotient H → quotient H :=
|
||||
quot.lift
|
||||
(λ a, ⟦a⁻¹⟧)
|
||||
(λ a₁ a₂ e, quot.sound (lcoset_equiv_inv H e))
|
||||
|
||||
private proposition qmul_assoc (a b c : quotient H) :
|
||||
qmul H (qmul H a b) c = qmul H a (qmul H b c) :=
|
||||
quot.induction_on₂ a b (λ a b, quot.induction_on c (λ c,
|
||||
have H : ⟦a * b * c⟧ = ⟦a * (b * c)⟧, by rewrite mul.assoc,
|
||||
H))
|
||||
|
||||
private proposition qmul_qone (a : quotient H) : qmul H a (qone H) = a :=
|
||||
quot.induction_on a (λ a', show ⟦a' * 1⟧ = ⟦a'⟧, by rewrite mul_one)
|
||||
|
||||
private proposition qone_qmul (a : quotient H) : qmul H (qone H) a = a :=
|
||||
quot.induction_on a (λ a', show ⟦1 * a'⟧ = ⟦a'⟧, by rewrite one_mul)
|
||||
|
||||
private proposition qmul_left_inv (a : quotient H) : qmul H (qinv H a) a = qone H :=
|
||||
quot.induction_on a (λ a', show ⟦a'⁻¹ * a'⟧ = ⟦1⟧, by rewrite mul.left_inv)
|
||||
|
||||
protected definition group [instance] : group (quotient H) :=
|
||||
⦃ group,
|
||||
mul := qmul H,
|
||||
inv := qinv H,
|
||||
one := qone H,
|
||||
mul_assoc := qmul_assoc H,
|
||||
mul_one := qmul_qone H,
|
||||
one_mul := qone_qmul H,
|
||||
mul_left_inv := qmul_left_inv H
|
||||
⦄
|
||||
|
||||
-- these theorems characterize the quotient group
|
||||
|
||||
definition qproj (a : A) : quotient H := ⟦a⟧
|
||||
|
||||
infix ` '* `:65 := λ {A' : Type} [group A'] a H' [is_normal H'], qproj H' a
|
||||
infix ` / ` := λ {A' : Type} [group A'] G H' [is_normal H'], qproj H' ' G
|
||||
|
||||
proposition is_hom_qproj [instance] : is_hom (qproj H) :=
|
||||
is_mul_hom.mk (λ a b, rfl)
|
||||
|
||||
variable {H}
|
||||
|
||||
proposition qproj_eq_qproj {a b : A} (h : a * H = b * H) : a '* H = b '* H :=
|
||||
quot.sound h
|
||||
|
||||
proposition lcoset_eq_lcoset_of_qproj_eq_qproj {a b : A} (h : a '* H = b '* H) : a * H = b * H :=
|
||||
quot.exact h
|
||||
|
||||
variable (H)
|
||||
|
||||
proposition qproj_eq_qproj_iff (a b : A) : a '* H = b '* H ↔ a * H = b * H :=
|
||||
iff.intro lcoset_eq_lcoset_of_qproj_eq_qproj qproj_eq_qproj
|
||||
|
||||
proposition ker_qproj [is_subgroup H] : ker (qproj H) = H :=
|
||||
ext (take a,
|
||||
begin
|
||||
rewrite [↑ker, mem_set_of_iff, -hom_one (qproj H), qproj_eq_qproj_iff,
|
||||
one_lcoset],
|
||||
show a * H = H ↔ a ∈ H, from iff.intro mem_of_lcoset_eq_self lcoset_eq_self_of_mem
|
||||
end)
|
||||
|
||||
proposition qproj_eq_one_iff [is_subgroup H] (a : A) : a '* H = 1 ↔ a ∈ H :=
|
||||
have H : qproj H a = 1 ↔ a ∈ ker (qproj H), from iff.rfl,
|
||||
by rewrite [H, ker_qproj]
|
||||
|
||||
variable {H}
|
||||
|
||||
proposition qproj_eq_one_of_mem [is_subgroup H] {a : A} (aH : a ∈ H) : a '* H = 1 :=
|
||||
iff.mpr (qproj_eq_one_iff H a) aH
|
||||
|
||||
proposition mem_of_qproj_eq_one [is_subgroup H] {a : A} (h : a '* H = 1) : a ∈ H :=
|
||||
iff.mp (qproj_eq_one_iff H a) h
|
||||
|
||||
variable (H)
|
||||
|
||||
proposition surjective_qproj : surjective (qproj H) :=
|
||||
take y, quot.induction_on y (λ a, exists.intro a rfl)
|
||||
|
||||
variable {H}
|
||||
|
||||
proposition quotient_induction {P : quotient H → Prop} (h : ∀ a, P (a '* H)) : ∀ a, P a :=
|
||||
surjective_induction (surjective_qproj H) h
|
||||
|
||||
proposition quotient_induction₂ {P : quotient H → quotient H → Prop}
|
||||
(h : ∀ a₁ a₂, P (a₁ '* H) (a₂ '* H)) :
|
||||
∀ a₁ a₂, P a₁ a₂ :=
|
||||
surjective_induction₂ (surjective_qproj H) h
|
||||
|
||||
variable (H)
|
||||
|
||||
proposition image_qproj_self [is_subgroup H] : H / H = '{1} :=
|
||||
eq_of_subset_of_subset
|
||||
(image_subset_of_maps_to
|
||||
(take x, suppose x ∈ H,
|
||||
show x '* H ∈ '{1},
|
||||
from mem_singleton_of_eq (qproj_eq_one_of_mem `x ∈ H`)))
|
||||
(take x, suppose x ∈ '{1},
|
||||
have x = 1, from eq_of_mem_singleton this,
|
||||
show x ∈ H / H, by rewrite this; apply mem_image_of_mem _ one_mem)
|
||||
|
||||
-- extending a function A → B to a function A / H → B
|
||||
|
||||
section respf
|
||||
|
||||
variable {H}
|
||||
variables {f : A → B} (respf : ∀ a₁ a₂, a₁ * H = a₂ * H → f a₁ = f a₂)
|
||||
|
||||
definition extend : quotient H → B := quot.lift f respf
|
||||
|
||||
proposition extend_qproj (a : A) : extend respf (a '* H) = f a := rfl
|
||||
|
||||
proposition extend_comp_qproj : extend respf ∘ (qproj H) = f := rfl
|
||||
|
||||
proposition image_extend (G : set A) : (extend respf) ' (G / H) = f ' G :=
|
||||
by rewrite [-image_comp]
|
||||
|
||||
variable [group B]
|
||||
|
||||
proposition is_hom_extend [instance] [is_hom f] : is_hom (extend respf) :=
|
||||
is_mul_hom.mk (take a b,
|
||||
show (extend respf (a * b)) = (extend respf a) * (extend respf b), from
|
||||
quot.induction_on₂ a b (take a b, hom_mul f a b))
|
||||
|
||||
proposition ker_extend : ker (extend respf) = ker f / H :=
|
||||
eq_of_subset_of_subset
|
||||
(quotient_induction
|
||||
(take a, assume Ha : qproj H a ∈ ker (extend respf),
|
||||
have f a = 1, from Ha,
|
||||
show a '* H ∈ ker f / H,
|
||||
from mem_image_of_mem _ this))
|
||||
(image_subset_of_maps_to
|
||||
(take a, assume h : a ∈ ker f,
|
||||
show extend respf (a '* H) = 1, from h))
|
||||
|
||||
end respf
|
||||
|
||||
end quotient_group
|
||||
|
||||
|
||||
/- the first homomorphism theorem for the quotient group -/
|
||||
|
||||
namespace quotient_group
|
||||
variables [group A] [group B] (f : A → B) [is_hom f]
|
||||
|
||||
lemma eq_of_lcoset_equiv_ker ⦃a b : A⦄ (h : lcoset_equiv (ker f) a b) : f a = f b :=
|
||||
have b⁻¹ * a ∈ ker f, from inv_mul_mem_of_lcoset_eq_lcoset h,
|
||||
eq.symm (eq_of_inv_mul_mem_ker this)
|
||||
|
||||
definition bar : quotient (ker f) → B := extend (eq_of_lcoset_equiv_ker f)
|
||||
|
||||
proposition bar_qproj (a : A) : bar f (a '* ker f) = f a := rfl
|
||||
|
||||
proposition is_hom_bar [instance] : is_hom (bar f) := is_hom_extend _
|
||||
|
||||
proposition image_bar (G : set A) : bar f ' (G / ker f) = f ' G :=
|
||||
by rewrite [↑bar, image_extend]
|
||||
|
||||
proposition image_bar_univ : bar f ' univ = f ' univ :=
|
||||
by rewrite [↑bar, -image_eq_univ_of_surjective (surjective_qproj (ker f)),
|
||||
image_extend]
|
||||
|
||||
proposition surj_on_bar : surj_on (bar f) univ (f ' univ) :=
|
||||
by rewrite [↑surj_on, image_bar_univ]; apply subset.refl
|
||||
|
||||
proposition ker_bar_eq : ker (bar f) = '{1} :=
|
||||
by rewrite [↑bar, ker_extend, image_qproj_self]
|
||||
|
||||
proposition injective_bar : injective (bar f) :=
|
||||
injective_of_ker_eq_singleton_one (ker_bar_eq f)
|
||||
end quotient_group
|
||||
|
||||
|
||||
/- a generic morphism extension property -/
|
||||
|
||||
section
|
||||
variables [group A] [group B] [group C]
|
||||
variables (G : set A) [is_subgroup G]
|
||||
variables (g : A → C) (f : A → B)
|
||||
|
||||
noncomputable definition gen_extend : C → B := λ c, f (inv_fun g G 1 c)
|
||||
|
||||
variables {G g f}
|
||||
|
||||
proposition eq_of_ker_in_subset {a₁ a₂ : A} (a₁G : a₁ ∈ G) (a₂G : a₂ ∈ G)
|
||||
[is_hom_on g G] [is_hom_on f G] (Hker : ker_in g G ⊆ ker f) (H' : g a₁ = g a₂) :
|
||||
f a₁ = f a₂ :=
|
||||
have memG : a₁⁻¹ * a₂ ∈ G, from mul_mem (inv_mem a₁G) a₂G,
|
||||
have a₁⁻¹ * a₂ ∈ ker_in g G, from inv_mul_mem_ker_in_of_eq a₁G a₂G H',
|
||||
have a₁⁻¹ * a₂ ∈ ker_in f G, from and.intro (Hker this) memG,
|
||||
show f a₁ = f a₂, from eq_of_inv_mul_mem_ker_in a₁G a₂G this
|
||||
|
||||
proposition gen_extend_spec [is_hom_on g G] [is_hom_on f G] (Hker : ker_in g G ⊆ ker f)
|
||||
{a : A} (aG : a ∈ G) : gen_extend G g f (g a) = f a :=
|
||||
eq_of_ker_in_subset (inv_fun_spec' aG) aG Hker (inv_fun_spec aG)
|
||||
|
||||
proposition is_hom_on_gen_extend [is_hom_on g G] [is_hom_on f G] (Hker : ker_in g G ⊆ ker f) :
|
||||
is_hom_on (gen_extend G g f) (g ' G) :=
|
||||
have is_subgroup (g ' G), from is_subgroup_image g G,
|
||||
take c₁, assume c₁gG : c₁ ∈ g ' G,
|
||||
take c₂, assume c₂gG : c₂ ∈ g ' G,
|
||||
let ginv := inv_fun g G 1 in
|
||||
have Hginv : maps_to ginv (g ' G) G, from maps_to_inv_fun one_mem,
|
||||
have ginvc₁ : ginv c₁ ∈ G, from Hginv c₁gG,
|
||||
have ginvc₂ : ginv c₂ ∈ G, from Hginv c₂gG,
|
||||
have ginvc₁c₂ : ginv (c₁ * c₂) ∈ G, from Hginv (mul_mem c₁gG c₂gG),
|
||||
have HH : ∀₀ c ∈ g ' G, g (ginv c) = c,
|
||||
from λ a aG, right_inv_on_inv_fun_of_surj_on _ (surj_on_image g G) aG,
|
||||
have eq₁ : g (ginv c₁) = c₁, from HH c₁gG,
|
||||
have eq₂ : g (ginv c₂) = c₂, from HH c₂gG,
|
||||
have eq₃ : g (ginv (c₁ * c₂)) = c₁ * c₂, from HH (mul_mem c₁gG c₂gG),
|
||||
have g (ginv (c₁ * c₂)) = g ((ginv c₁) * (ginv c₂)),
|
||||
by rewrite [eq₃, hom_on_mul g ginvc₁ ginvc₂, eq₁, eq₂],
|
||||
have f (ginv (c₁ * c₂)) = f (ginv c₁ * ginv c₂),
|
||||
from eq_of_ker_in_subset (ginvc₁c₂) (mul_mem ginvc₁ ginvc₂) Hker this,
|
||||
show f (ginv (c₁ * c₂)) = f (ginv c₁) * f (ginv c₂),
|
||||
by rewrite [this, hom_on_mul f ginvc₁ ginvc₂]
|
||||
end
|
||||
|
||||
|
||||
/- quotient by an arbitrary group, not necessarily normal -/
|
||||
|
||||
namespace quotient_group_general
|
||||
|
||||
variables [group A] (H : set A) [is_subgroup H]
|
||||
|
||||
lemma is_normal_to_group_of_normalizer [instance] :
|
||||
is_normal (to_group_of (normalizer H) ' H) :=
|
||||
have H1 : is_normal_in (to_group_of (normalizer H) ' H)
|
||||
(to_group_of (normalizer H) ' (normalizer H)),
|
||||
from is_normal_in_image_image (subset_normalizer_self H) (to_group_of (normalizer H)),
|
||||
have H2 : to_group_of (normalizer H) ' (normalizer H) = univ,
|
||||
from image_to_group_of_eq_univ (normalizer H),
|
||||
is_normal_of_is_normal_in_univ (by rewrite -H2; exact H1)
|
||||
|
||||
section quotient_group
|
||||
open quotient_group
|
||||
|
||||
noncomputable definition quotient : Type := quotient (to_group_of (normalizer H) ' H)
|
||||
|
||||
noncomputable definition group_quotient [instance] : group (quotient H) :=
|
||||
quotient_group.group (to_group_of (normalizer H) ' H)
|
||||
|
||||
noncomputable definition qproj : A → quotient H :=
|
||||
qproj (to_group_of (normalizer H) ' H) ∘ (to_group_of (normalizer H))
|
||||
|
||||
infix ` '* `:65 := λ {A' : Type} [group A'] a H' [is_subgroup H'], qproj H' a
|
||||
infix ` / ` := λ {A' : Type} [group A'] G H' [is_subgroup H'], qproj H' ' G
|
||||
|
||||
proposition is_hom_on_qproj [instance] : is_hom_on (qproj H) (normalizer H) :=
|
||||
have H₀ : is_hom_on (to_group_of (normalizer H)) (normalizer H),
|
||||
from is_hom_on_to_group_of (normalizer H),
|
||||
have H₁ : is_hom_on (quotient_group.qproj (to_group_of (normalizer H) ' H)) univ,
|
||||
from iff.mpr (is_hom_on_univ_iff (quotient_group.qproj (to_group_of (normalizer H) ' H)))
|
||||
(is_hom_qproj (to_group_of (normalizer H) ' H)),
|
||||
is_hom_on_comp H₀ H₁ (maps_to_univ (to_group_of (normalizer H)) (normalizer H))
|
||||
|
||||
proposition is_hom_on_qproj' [instance] (G : set A) [is_normal_in H G] :
|
||||
is_hom_on (qproj H) G :=
|
||||
is_hom_on_of_subset (qproj H) (subset_normalizer G H)
|
||||
|
||||
proposition ker_in_qproj : ker_in (qproj H) (normalizer H) = H :=
|
||||
let tg := to_group_of (normalizer H) in
|
||||
begin
|
||||
rewrite [↑ker_in, ker_eq_preimage_one, ↑qproj, preimage_comp, -ker_eq_preimage_one],
|
||||
have is_hom_on tg H, from is_hom_on_of_subset _ (subset_normalizer_self H),
|
||||
have is_subgroup (tg ' H), from is_subgroup_image tg H,
|
||||
krewrite [ker_qproj, to_group_of_preimage_to_group_of_image (subset_normalizer_self H)]
|
||||
end
|
||||
|
||||
end quotient_group
|
||||
|
||||
variable {H}
|
||||
|
||||
proposition qproj_eq_qproj_iff {a b : A} (Ha : a ∈ normalizer H) (Hb : b ∈ normalizer H) :
|
||||
a '* H = b '* H ↔ a * H = b * H :=
|
||||
by rewrite [lcoset_eq_lcoset_iff, eq_iff_inv_mul_mem_ker_in Ha Hb, ker_in_qproj,
|
||||
-inv_mem_iff, mul_inv, inv_inv]
|
||||
|
||||
proposition qproj_eq_qproj {a b : A} (Ha : a ∈ normalizer H) (Hb : b ∈ normalizer H)
|
||||
(h : a * H = b * H) :
|
||||
a '* H = b '* H :=
|
||||
iff.mpr (qproj_eq_qproj_iff Ha Hb) h
|
||||
|
||||
proposition lcoset_eq_lcoset_of_qproj_eq_qproj {a b : A}
|
||||
(Ha : a ∈ normalizer H) (Hb : b ∈ normalizer H) (h : a '* H = b '* H) :
|
||||
a * H = b * H :=
|
||||
iff.mp (qproj_eq_qproj_iff Ha Hb) h
|
||||
|
||||
variable (H)
|
||||
|
||||
proposition qproj_mem {a : A} {G : set A} (aG : a ∈ G) : a '* H ∈ G / H :=
|
||||
mem_image_of_mem _ aG
|
||||
|
||||
proposition qproj_one : 1 '* H = 1 := hom_on_one (qproj H) (normalizer H)
|
||||
|
||||
variable {H}
|
||||
|
||||
proposition mem_of_qproj_mem {a : A} (anH : a ∈ normalizer H)
|
||||
{G : set A} (HsubG : H ⊆ G) [is_subgroup G] [is_normal_in H G]
|
||||
(aHGH : a '* H ∈ G / H): a ∈ G :=
|
||||
have GH : G ⊆ normalizer H, from subset_normalizer G H,
|
||||
obtain b [bG (bHeq : b '* H = a '* H)], from aHGH,
|
||||
have b * H = a * H, from lcoset_eq_lcoset_of_qproj_eq_qproj (GH bG) anH bHeq,
|
||||
have a ∈ b * H, by rewrite this; apply mem_lcoset_self,
|
||||
have a ∈ b * G, from lcoset_subset_lcoset b HsubG this,
|
||||
show a ∈ G, by rewrite [lcoset_eq_self_of_mem bG at this]; apply this
|
||||
|
||||
proposition qproj_eq_one_iff {a : A} (Ha : a ∈ normalizer H) : a '* H = 1 ↔ a ∈ H :=
|
||||
by rewrite [-hom_on_one (qproj H) (normalizer H), qproj_eq_qproj_iff Ha one_mem, one_lcoset,
|
||||
lcoset_eq_self_iff]
|
||||
|
||||
proposition qproj_eq_one_of_mem {a : A} (aH : a ∈ H) : a '* H = 1 :=
|
||||
iff.mpr (qproj_eq_one_iff (subset_normalizer_self H aH)) aH
|
||||
|
||||
proposition mem_of_qproj_eq_one {a : A} (Ha : a ∈ normalizer H) (h : a '* H = 1) : a ∈ H :=
|
||||
iff.mp (qproj_eq_one_iff Ha) h
|
||||
|
||||
variable (H)
|
||||
|
||||
section
|
||||
open quotient_group
|
||||
proposition surj_on_qproj_normalizer : surj_on (qproj H) (normalizer H) univ :=
|
||||
have H₀ : surj_on (to_group_of (normalizer H)) (normalizer H) univ,
|
||||
from surj_on_to_group_of_univ (normalizer H),
|
||||
have H₁ : surj_on (quotient_group.qproj (to_group_of (normalizer H) ' H)) univ univ,
|
||||
from surj_on_univ_of_surjective univ (surjective_qproj _),
|
||||
surj_on_comp H₁ H₀
|
||||
end
|
||||
|
||||
variable {H}
|
||||
|
||||
proposition quotient_induction {P : quotient H → Prop} (hyp : ∀₀ a ∈ normalizer H, P (a '* H)) :
|
||||
∀ a, P a :=
|
||||
surj_on_univ_induction (surj_on_qproj_normalizer H) hyp
|
||||
|
||||
proposition quotient_induction₂ {P : quotient H → quotient H → Prop}
|
||||
(hyp : ∀₀ a₁ ∈ normalizer H, ∀₀ a₂ ∈ normalizer H, P (a₁ '* H) (a₂ '* H)) :
|
||||
∀ a₁ a₂, P a₁ a₂ :=
|
||||
surj_on_univ_induction₂ (surj_on_qproj_normalizer H) hyp
|
||||
|
||||
variable (H)
|
||||
|
||||
proposition image_qproj_self : H / H = '{1} :=
|
||||
eq_of_subset_of_subset
|
||||
(image_subset_of_maps_to
|
||||
(take x, suppose x ∈ H,
|
||||
show x '* H ∈ '{1},
|
||||
from mem_singleton_of_eq (qproj_eq_one_of_mem `x ∈ H`)))
|
||||
(take x, suppose x ∈ '{1},
|
||||
have x = 1, from eq_of_mem_singleton this,
|
||||
show x ∈ H / H,
|
||||
by rewrite [this, -qproj_one H]; apply mem_image_of_mem _ one_mem)
|
||||
|
||||
section respf
|
||||
|
||||
variable (H)
|
||||
variables [group B] (G : set A) [is_subgroup G] (f : A → B)
|
||||
|
||||
noncomputable definition extend : quotient H → B := gen_extend G (qproj H) f
|
||||
|
||||
variables [is_hom_on f G] [is_normal_in H G]
|
||||
|
||||
private proposition aux : is_hom_on (qproj H) G :=
|
||||
is_hom_on_of_subset (qproj H) (subset_normalizer G H)
|
||||
|
||||
local attribute [instance] aux
|
||||
|
||||
variables {H f}
|
||||
|
||||
private proposition aux' (respf : H ⊆ ker f) : ker_in (qproj H) G ⊆ ker f :=
|
||||
subset.trans
|
||||
(show ker_in (qproj H) G ⊆ ker_in (qproj H) (normalizer H),
|
||||
from inter_subset_inter_left _ (subset_normalizer G H))
|
||||
(by rewrite [ker_in_qproj]; apply respf)
|
||||
|
||||
variable {G}
|
||||
|
||||
proposition extend_qproj (respf : H ⊆ ker f) {a : A} (aG : a ∈ G) :
|
||||
extend H G f (a '* H) = f a :=
|
||||
gen_extend_spec (aux' G respf) aG
|
||||
|
||||
proposition image_extend (respf : H ⊆ ker f) {s : set A} (ssubG : s ⊆ G) :
|
||||
extend H G f ' (s / H) = f ' s :=
|
||||
begin
|
||||
rewrite [-image_comp],
|
||||
apply image_eq_image_of_eq_on,
|
||||
intro a amems,
|
||||
apply extend_qproj respf (ssubG amems)
|
||||
end
|
||||
|
||||
variable (G)
|
||||
|
||||
proposition is_hom_on_extend [instance] (respf : H ⊆ ker f) : is_hom_on (extend H G f) (G / H) :=
|
||||
by unfold extend; apply is_hom_on_gen_extend (aux' G respf)
|
||||
|
||||
variable {G}
|
||||
|
||||
proposition ker_in_extend [is_subgroup G] (respf : H ⊆ ker f) (HsubG : H ⊆ G) :
|
||||
ker_in (extend H G f) (G / H) = (ker_in f G) / H :=
|
||||
begin
|
||||
apply ext,
|
||||
intro aH,
|
||||
cases surj_on_qproj_normalizer H (show aH ∈ univ, from trivial) with a atemp,
|
||||
cases atemp with anH aHeq,
|
||||
rewrite -aHeq,
|
||||
apply iff.intro,
|
||||
{ intro akerin,
|
||||
cases akerin with aker ain,
|
||||
have a '* H ∈ G / H, from ain,
|
||||
have a ∈ G, from mem_of_qproj_mem anH HsubG this,
|
||||
have a '* H ∈ ker (extend H G f), from aker,
|
||||
have extend H G f (a '* H) = 1, from this,
|
||||
have f a = extend H G f (a '* H), from eq.symm (extend_qproj respf `a ∈ G`),
|
||||
have f a = 1, by rewrite this; assumption,
|
||||
have a ∈ ker_in f G, from and.intro this `a ∈ G`,
|
||||
show a '* H ∈ (ker_in f G) / H, from qproj_mem H this},
|
||||
intro aHker,
|
||||
have aker : a ∈ ker_in f G,
|
||||
begin
|
||||
have Hsub : H ⊆ ker_in f G, from subset_inter respf HsubG,
|
||||
have is_normal_in H (ker_in f G),
|
||||
from subset.trans (inter_subset_right (ker f) G) (subset_normalizer G H),
|
||||
apply (mem_of_qproj_mem anH Hsub aHker)
|
||||
end,
|
||||
have a ∈ G, from and.right aker,
|
||||
have f a = 1, from and.left aker,
|
||||
have extend H G f (a '* H) = 1,
|
||||
from eq.trans (extend_qproj respf `a ∈ G`) this,
|
||||
show a '* H ∈ ker_in (extend H G f) (G / H),
|
||||
from and.intro this (qproj_mem H `a ∈ G`)
|
||||
end
|
||||
|
||||
/- (comment from Jeremy)
|
||||
This version kills the elaborator. I don't know why.
|
||||
Tracing class instances doesn't show a problem. My best guess is that it is
|
||||
the backgracking from the "obtain".
|
||||
|
||||
proposition ker_in_extend [is_subgroup G] (respf : H ⊆ ker f) (HsubG : H ⊆ G) :
|
||||
ker_in (extend H G f) (qproj H ' G) = qproj H ' (ker_in f G) :=
|
||||
ext (take aH,
|
||||
obtain a [(anH : a ∈ normalizer H) (aHeq : a '* H = aH)],
|
||||
from surj_on_qproj_normalizer H (show aH ∈ univ, from trivial),
|
||||
begin
|
||||
rewrite -aHeq, apply iff.intro, unfold ker_in,
|
||||
exact
|
||||
(assume aker : a '* H ∈ ker (extend H G f) ∩ (qproj H ' G),
|
||||
have a '* H ∈ qproj H ' G, from and.right aker,
|
||||
have a ∈ G, from mem_of_qproj_mem anH HsubG this,
|
||||
-- Uncommenting the next line of code slows things down dramatically.
|
||||
-- Uncommenting the one after kills the system.
|
||||
-- have a '* H ∈ ker (extend H G f), from and.left aker,
|
||||
-- have extend H G f (a '* H) = 1, from this,
|
||||
-- have f a = extend H G f (a '* H), from eq.symm (extend_qproj respf `a ∈ G`),
|
||||
-- have f a = 1, by rewrite [-this, extend_qproj respf aG],
|
||||
-- have a ∈ ker_in f G, from and.intro this `a ∈ G`,
|
||||
show a '* H ∈ qproj H ' (ker_in f G), from sorry),
|
||||
exact
|
||||
(assume hyp : a '* H ∈ qproj H ' (ker_in f G),
|
||||
show a '* H ∈ ker_in (extend H G f) (qproj H ' G), from sorry)
|
||||
end)
|
||||
-/
|
||||
|
||||
end respf
|
||||
|
||||
attribute quotient [irreducible]
|
||||
|
||||
end quotient_group_general
|
||||
|
||||
/- the first homomorphism theorem for general quotient groups -/
|
||||
|
||||
namespace quotient_group_general
|
||||
|
||||
variables [group A] [group B] (G : set A) [is_subgroup G]
|
||||
variables (f : A → B) [is_hom_on f G]
|
||||
|
||||
noncomputable definition bar : quotient (ker_in f G) → B :=
|
||||
extend (ker_in f G) G f
|
||||
|
||||
proposition bar_qproj {a : A} (aG : a ∈ G) : bar G f (a '* ker_in f G) = f a :=
|
||||
extend_qproj (inter_subset_left _ _) aG
|
||||
|
||||
proposition is_hom_on_bar [instance] : is_hom_on (bar G f) (G / ker_in f G) :=
|
||||
have is_subgroup (ker f ∩ G), from is_subgroup_ker_in f G,
|
||||
have is_normal_in (ker f ∩ G) G, from is_normal_in_ker_in f G,
|
||||
is_hom_on_extend G (inter_subset_left _ _)
|
||||
|
||||
proposition image_bar {s : set A} (ssubG : s ⊆ G) : bar G f ' (s / ker_in f G) = f ' s :=
|
||||
have is_subgroup (ker f ∩ G), from is_subgroup_ker_in f G,
|
||||
have is_normal_in (ker f ∩ G) G, from is_normal_in_ker_in f G,
|
||||
image_extend (inter_subset_left _ _) ssubG
|
||||
|
||||
proposition surj_on_bar : surj_on (bar G f) (G / ker_in f G) (f ' G) :=
|
||||
by rewrite [↑surj_on, image_bar G f (@subset.refl _ G)]; apply subset.refl
|
||||
|
||||
proposition ker_in_bar : ker_in (bar G f) (G / ker_in f G) = '{1} :=
|
||||
have H₀ : ker_in f G ⊆ ker f, from inter_subset_left _ _,
|
||||
have H₁ : ker_in f G ⊆ G, from inter_subset_right _ _,
|
||||
by rewrite [↑bar, ker_in_extend H₀ H₁, image_qproj_self]
|
||||
|
||||
proposition inj_on_bar : inj_on (bar G f) (G / ker_in f G) :=
|
||||
inj_on_of_ker_in_eq_singleton_one (ker_in_bar G f)
|
||||
|
||||
end quotient_group_general
|
||||
|
||||
end group_theory
|
||||
99
library/theories/group_theory/subgroup_to_group.lean
Normal file
99
library/theories/group_theory/subgroup_to_group.lean
Normal file
|
|
@ -0,0 +1,99 @@
|
|||
/-
|
||||
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Jeremy Avigad
|
||||
|
||||
Turn a subgroup into a group on the corresponding subtype. Given
|
||||
|
||||
variables {A : Type} [group A] (G : set A) [is_subgroup G]
|
||||
|
||||
we have:
|
||||
|
||||
group_of G := G, viewed as a group
|
||||
to_group_of G a := if a is in G, returns the image in group_of G, or 1 otherwise
|
||||
to_subgroup a := given a : group_of G, return the underlying element
|
||||
-/
|
||||
import .basic
|
||||
open set function subtype classical
|
||||
|
||||
variables {A B C : Type}
|
||||
|
||||
namespace group_theory
|
||||
|
||||
definition group_of (G : set A) : Type := subtype G
|
||||
|
||||
definition subgroup_to_group {G : set A} ⦃a : A⦄ (aG : a ∈ G) : group_of G := tag a aG
|
||||
|
||||
definition to_subgroup {G : set A} (a : group_of G) : A := elt_of a
|
||||
|
||||
proposition to_subgroup_mem {G : set A} (a : group_of G) : to_subgroup a ∈ G := has_property a
|
||||
|
||||
variables [group A] (G : set A) [is_subgroup G]
|
||||
|
||||
definition group_of.group [instance] : group (group_of G) :=
|
||||
⦃ group,
|
||||
mul := λ a b, subgroup_to_group (mul_mem (to_subgroup_mem a) (to_subgroup_mem b)),
|
||||
mul_assoc := λ a b c, subtype.eq !mul.assoc,
|
||||
one := subgroup_to_group (@one_mem A _ G _),
|
||||
one_mul := λ a, subtype.eq !one_mul,
|
||||
mul_one := λ a, subtype.eq !mul_one,
|
||||
inv := λ a, tag (elt_of a)⁻¹ (inv_mem (to_subgroup_mem a)),
|
||||
mul_left_inv := λ a, subtype.eq !mul.left_inv
|
||||
⦄
|
||||
|
||||
proposition is_hom_group_to_subgroup [instance] : is_hom (@to_subgroup A G) :=
|
||||
is_mul_hom.mk
|
||||
(take g₁ g₂ : group_of G,
|
||||
show to_subgroup (g₁ * g₂) = to_subgroup g₁ * to_subgroup g₂,
|
||||
by cases g₁; cases g₂; reflexivity)
|
||||
|
||||
noncomputable definition to_group_of (a : A) : group_of G :=
|
||||
if H : a ∈ G then subgroup_to_group H else 1
|
||||
|
||||
proposition is_hom_on_to_group_of [instance] : is_hom_on (to_group_of G) G :=
|
||||
take g₁, assume g₁G, take g₂, assume g₂G,
|
||||
show to_group_of G (g₁ * g₂) = to_group_of G g₁ * to_group_of G g₂,
|
||||
by rewrite [↑to_group_of, dif_pos g₁G, dif_pos g₂G, dif_pos (mul_mem g₁G g₂G)]
|
||||
|
||||
proposition to_group_to_subgroup : left_inverse (to_group_of G) to_subgroup :=
|
||||
begin
|
||||
intro a, rewrite [↑to_group_of, dif_pos (to_subgroup_mem a)],
|
||||
apply subtype.eq, reflexivity
|
||||
end
|
||||
|
||||
-- proposition to_subgroup_to_group {a : A} (aG : a ∈ G) : to_subgroup (to_group_of G a) = a :=
|
||||
-- by rewrite [↑to_group_of, dif_pos aG]
|
||||
-- curiously, in the next version, "by rewrite [↑to_group_of, dif_pos aG]" doesn't work.
|
||||
|
||||
proposition to_subgroup_to_group : left_inv_on to_subgroup (to_group_of G) G :=
|
||||
λ a aG, by xrewrite [dif_pos aG]
|
||||
|
||||
variable {G}
|
||||
proposition inj_on_to_group_of : inj_on (to_group_of G) G :=
|
||||
inj_on_of_left_inv_on (to_subgroup_to_group G)
|
||||
variable (G)
|
||||
|
||||
proposition surj_on_to_group_of_univ : surj_on (to_group_of G) G univ :=
|
||||
take y, assume yuniv, mem_image (to_subgroup_mem y) (to_group_to_subgroup G y)
|
||||
|
||||
proposition image_to_group_of_eq_univ : to_group_of G ' G = univ :=
|
||||
image_eq_of_maps_to_of_surj_on (maps_to_univ _ _) (surj_on_to_group_of_univ G)
|
||||
|
||||
proposition surjective_to_group_of : surjective (to_group_of G) :=
|
||||
surjective_of_has_right_inverse (exists.intro _ (to_group_to_subgroup G))
|
||||
|
||||
variable {G}
|
||||
proposition to_group_of_preimage_to_group_of_image {S : set A} (SsubG : S ⊆ G) :
|
||||
(to_group_of G) '- (to_group_of G ' S) ∩ G = S :=
|
||||
ext (take x, iff.intro
|
||||
(assume H,
|
||||
obtain Hx (xG : x ∈ G), from H,
|
||||
have to_group_of G x ∈ to_group_of G ' S, from mem_of_mem_preimage Hx,
|
||||
obtain y [(yS : y ∈ S) (Heq : to_group_of G y = to_group_of G x)], from this,
|
||||
have y = x, from inj_on_to_group_of (SsubG yS) xG Heq,
|
||||
show x ∈ S, by rewrite -this; exact yS)
|
||||
(assume xS, and.intro
|
||||
(mem_preimage (show to_group_of G x ∈ to_group_of G ' S, from mem_image_of_mem _ xS))
|
||||
(SsubG xS)))
|
||||
|
||||
end group_theory
|
||||
|
|
@ -7,9 +7,46 @@ Temporary file; move in Lean3.
|
|||
import data.set algebra.order_bigops
|
||||
import data.finset data.list.sort
|
||||
|
||||
-- move to data.set
|
||||
-- move this to init.function
|
||||
|
||||
section
|
||||
open function
|
||||
postfix `^~` :std.prec.max_plus := swap
|
||||
end
|
||||
|
||||
-- move to algebra
|
||||
|
||||
theorem eq_of_inv_mul_eq_one {A : Type} {a b : A} [group A] (H : b⁻¹ * a = 1) : a = b :=
|
||||
have a⁻¹ * 1 = a⁻¹, by inst_simp,
|
||||
by inst_simp
|
||||
|
||||
-- move to init.quotient
|
||||
|
||||
namespace quot
|
||||
open classical
|
||||
|
||||
variables {A : Type} [s : setoid A]
|
||||
|
||||
protected theorem exists_eq_mk (x : quot s) : ∃ a : A, x = ⟦a⟧ :=
|
||||
quot.induction_on x (take a, exists.intro _ rfl)
|
||||
|
||||
protected noncomputable definition repr (x : quot s) : A := some (quot.exists_eq_mk x)
|
||||
|
||||
protected theorem mk_repr_eq (x : quot s) : ⟦ quot.repr x ⟧ = x :=
|
||||
eq.symm (some_spec (quot.exists_eq_mk x))
|
||||
|
||||
open setoid
|
||||
include s
|
||||
protected theorem repr_mk_equiv (a : A) : quot.repr ⟦a⟧ ≈ a :=
|
||||
quot.exact (by rewrite quot.mk_repr_eq)
|
||||
|
||||
end quot
|
||||
|
||||
|
||||
-- move to data.set.basic
|
||||
|
||||
namespace set
|
||||
open function
|
||||
|
||||
lemma inter_eq_self_of_subset {X : Type} {s t : set X} (Hst : s ⊆ t) : s ∩ t = s :=
|
||||
ext (take x, iff.intro
|
||||
|
|
@ -81,6 +118,146 @@ proposition preimage_Union {I X Y : Type} (f : X → Y) (u : I → set Y) :
|
|||
f '- (⋃ i, u i) = ⋃ i, (f '- (u i)) :=
|
||||
ext (take x, !iff.refl)
|
||||
|
||||
-- TODO: rename "injective" to "inj"
|
||||
-- TODO: turn around equality in definition of image
|
||||
-- TODO: use ∀₀ in definition of injective (and define notation for ∀₀ x y ∈ s, ...)
|
||||
|
||||
attribute [trans] subset.trans -- really? this was never declared? And all the variants...
|
||||
|
||||
proposition mem_set_of_iff {X : Type} (P : X → Prop) (a : X) : a ∈ { x : X | P x } ↔ P a :=
|
||||
iff.refl _
|
||||
|
||||
proposition mem_set_of {X : Type} {P : X → Prop} {a : X} (Pa : P a) : a ∈ { x : X | P x } := Pa
|
||||
|
||||
proposition of_mem_set_of {X : Type} {P : X → Prop} {a : X} (H : a ∈ { x : X | P x }) : P a := H
|
||||
|
||||
proposition forallb_of_forall {X : Type} {P : X → Prop} (s : set X) (H : ∀ x, P x) :
|
||||
∀₀ x ∈ s, P x :=
|
||||
λ x xs, H x
|
||||
|
||||
proposition forall_of_forallb_univ {X : Type} {P : X → Prop} (H : ∀₀ x ∈ univ, P x) : ∀ x, P x :=
|
||||
λ x, H trivial
|
||||
|
||||
proposition forallb_univ_iff_forall {X : Type} (P : X → Prop) : (∀₀ x ∈ univ, P x) ↔ ∀ x, P x :=
|
||||
iff.intro forall_of_forallb_univ !forallb_of_forall
|
||||
|
||||
proposition forallb_of_subset {X : Type} {s t : set X} {P : X → Prop}
|
||||
(ssubt : s ⊆ t) (Ht : ∀₀ x ∈ t, P x) : ∀₀ x ∈ s, P x :=
|
||||
λ x xs, Ht (ssubt xs)
|
||||
|
||||
proposition forallb_of_forall₂ {X Y : Type} {P : X → Y → Prop} (s : set X) (t : set Y)
|
||||
(H : ∀ x y, P x y) : ∀₀ x ∈ s, ∀₀ y ∈ t, P x y :=
|
||||
λ x xs y yt, H x y
|
||||
|
||||
proposition forall_of_forallb_univ₂ {X Y : Type} {P : X → Y → Prop}
|
||||
(H : ∀₀ x ∈ univ, ∀₀ y ∈ univ, P x y) : ∀ x y, P x y :=
|
||||
λ x y, H trivial trivial
|
||||
|
||||
proposition forallb_univ_iff_forall₂ {X Y : Type} (P : X → Y → Prop) :
|
||||
(∀₀ x ∈ univ, ∀₀ y ∈ univ, P x y) ↔ ∀ x y, P x y :=
|
||||
iff.intro forall_of_forallb_univ₂ !forallb_of_forall₂
|
||||
|
||||
proposition forallb_of_subset₂ {X Y : Type} {s₁ t₁ : set X} {s₂ t₂ : set Y} {P : X → Y → Prop}
|
||||
(ssubt₁ : s₁ ⊆ t₁) (ssubt₂ : s₂ ⊆ t₂) (Ht : ∀₀ x ∈ t₁, ∀₀ y ∈ t₂, P x y) :
|
||||
∀₀ x ∈ s₁, ∀₀ y ∈ s₂, P x y :=
|
||||
λ x xs y ys, Ht (ssubt₁ xs) (ssubt₂ ys)
|
||||
|
||||
theorem maps_to_univ {X Y : Type} (f : X → Y) (a : set X) : maps_to f a univ :=
|
||||
take x, assume H, trivial
|
||||
|
||||
theorem surj_on_image {X Y : Type} (f : X → Y) (a : set X) : surj_on f a (f ' a) :=
|
||||
λ y Hy, Hy
|
||||
|
||||
theorem image_eq_univ_of_surjective {X Y : Type} {f : X → Y} (H : surjective f) :
|
||||
f ' univ = univ :=
|
||||
ext (take y, iff.intro (λ H', trivial)
|
||||
(λ H', obtain x xeq, from H y,
|
||||
show y ∈ f ' univ, from mem_image trivial xeq))
|
||||
|
||||
proposition image_inter_subset {X Y : Type} (f : X → Y) (s t : set X) :
|
||||
f ' (s ∩ t) ⊆ f ' s ∩ f ' t :=
|
||||
take y, assume ymem,
|
||||
obtain x [[xs xt] (xeq : f x = y)], from ymem,
|
||||
show y ∈ f ' s ∩ f ' t,
|
||||
begin
|
||||
rewrite -xeq,
|
||||
exact (and.intro (mem_image_of_mem f xs) (mem_image_of_mem f xt))
|
||||
end
|
||||
|
||||
--proposition image_eq_of_maps_to_of_surj_on {X Y : Type} {f : X → Y} {s : set X} {t : set Y}
|
||||
-- (H : maps_to f s t) (H' : surj_on f s t) :
|
||||
-- f ' s = t :=
|
||||
--eq_of_subset_of_subset (image_subset_of_maps_to H) H'
|
||||
|
||||
proposition surj_on_of_image_eq {X Y : Type} {f : X → Y} {s : set X} {t : set Y}
|
||||
(H : f ' s = t) :
|
||||
surj_on f s t :=
|
||||
by rewrite [↑surj_on, H]; apply subset.refl
|
||||
|
||||
proposition surjective_induction {X Y : Type} {P : Y → Prop} {f : X → Y}
|
||||
(surjf : surjective f) (H : ∀ x, P (f x)) :
|
||||
∀ y, P y :=
|
||||
take y,
|
||||
obtain x (yeq : f x = y), from surjf y,
|
||||
show P y, by rewrite -yeq; apply H x
|
||||
|
||||
proposition surjective_induction₂ {X Y : Type} {P : Y → Y → Prop} {f : X → Y}
|
||||
(surjf : surjective f) (H : ∀ x₁ x₂, P (f x₁) (f x₂)) :
|
||||
∀ y₁ y₂, P y₁ y₂ :=
|
||||
take y₁ y₂,
|
||||
obtain x₁ (y₁eq : f x₁ = y₁), from surjf y₁,
|
||||
obtain x₂ (y₂eq : f x₂ = y₂), from surjf y₂,
|
||||
show P y₁ y₂, by rewrite [-y₁eq, -y₂eq]; apply H x₁ x₂
|
||||
|
||||
proposition surj_on_univ_induction {X Y : Type} {P : Y → Prop} {f : X → Y} {s : set X}
|
||||
(surjfs : surj_on f s univ) (H : ∀₀ x ∈ s, P (f x)) :
|
||||
∀ y, P y :=
|
||||
take y,
|
||||
obtain x [xs (yeq : f x = y)], from surjfs trivial,
|
||||
show P y, by rewrite -yeq; apply H xs
|
||||
|
||||
proposition surj_on_univ_induction₂ {X Y : Type} {P : Y → Y → Prop} {f : X → Y} {s : set X}
|
||||
(surjfs : surj_on f s univ) (H : ∀₀ x₁ ∈ s, ∀₀ x₂ ∈ s, P (f x₁) (f x₂)) :
|
||||
∀ y₁ y₂, P y₁ y₂ :=
|
||||
take y₁ y₂,
|
||||
obtain x₁ [x₁s (y₁eq : f x₁ = y₁)], from surjfs trivial,
|
||||
obtain x₂ [x₂s (y₂eq : f x₂ = y₂)], from surjfs trivial,
|
||||
show P y₁ y₂, by rewrite [-y₁eq, -y₂eq]; apply H x₁s x₂s
|
||||
|
||||
proposition surj_on_univ_of_surjective {X Y : Type} {f : X → Y} (s : set Y) (H : surjective f) :
|
||||
surj_on f univ s :=
|
||||
take y, assume ys,
|
||||
obtain x yeq, from H y,
|
||||
mem_image !mem_univ yeq
|
||||
|
||||
proposition mem_of_mem_image_of_injective {X Y : Type} {f : X → Y} {s : set X} {a : X}
|
||||
(injf : injective f) (H : f a ∈ f ' s) :
|
||||
a ∈ s :=
|
||||
obtain b [bs faeq], from H,
|
||||
have b = a, from injf faeq,
|
||||
by rewrite -this; apply bs
|
||||
|
||||
proposition mem_of_mem_image_of_inj_on {X Y : Type} {f : X → Y} {s t : set X} {a : X} (Ha : a ∈ t)
|
||||
(Hs : s ⊆ t) (injft : inj_on f t) (H : f a ∈ f ' s) :
|
||||
a ∈ s :=
|
||||
obtain b [bs faeq], from H,
|
||||
have b = a, from injft (Hs bs) Ha faeq,
|
||||
by rewrite -this; apply bs
|
||||
|
||||
proposition eq_singleton_of_forall_eq {A : Type} {s : set A} {x : A} (xs : x ∈ s) (H : ∀₀ y ∈ s, y = x) :
|
||||
s = '{x} :=
|
||||
ext (take y, iff.intro
|
||||
(assume ys, mem_singleton_of_eq (H ys))
|
||||
(assume yx, by rewrite (eq_of_mem_singleton yx); assumption))
|
||||
|
||||
proposition insert_subset {A : Type} {s t : set A} {a : A} (amem : a ∈ t) (ssubt : s ⊆ t) : insert a s ⊆ t :=
|
||||
take x, assume xias,
|
||||
or.elim (eq_or_mem_of_mem_insert xias)
|
||||
(by simp)
|
||||
(take H, ssubt H)
|
||||
|
||||
-- move to data.set.finite
|
||||
|
||||
lemma finite_sUnion {A : Type} {S : set (set A)} [H : finite S] :
|
||||
(∀s, s ∈ S → finite s) → finite ⋃₀S :=
|
||||
induction_on_finite S
|
||||
|
|
@ -100,6 +277,49 @@ lemma finite_of_finite_sUnion {A : Type} (S : set (set A)) (H : finite ⋃₀S)
|
|||
have finite (𝒫 (⋃₀ S)), from finite_powerset _,
|
||||
show finite S, from finite_subset (subset_powerset_sUnion S)
|
||||
|
||||
section nat
|
||||
open nat
|
||||
|
||||
proposition ne_empty_of_card_pos {A : Type} {s : set A} (H : card s > 0) : s ≠ ∅ :=
|
||||
take H', begin rewrite [H' at H, card_empty at H], exact lt.irrefl 0 H end
|
||||
|
||||
lemma eq_of_card_eq_one {A : Type} {S : set A} (H : card S = 1) {x y : A} (Hx : x ∈ S) (Hy : y ∈ S) :
|
||||
x = y :=
|
||||
have finite S,
|
||||
from classical.by_contradiction
|
||||
(assume nfinS, begin rewrite (card_of_not_finite nfinS) at H, contradiction end),
|
||||
classical.by_contradiction
|
||||
(assume H0 : x ≠ y,
|
||||
have H1 : '{x, y} ⊆ S, from insert_subset Hx (insert_subset Hy (empty_subset _)),
|
||||
have x ∉ '{y}, from assume H, H0 (eq_of_mem_singleton H),
|
||||
have 2 ≤ 1, from calc
|
||||
2 = card '{x, y} : by rewrite [card_insert_of_not_mem this,
|
||||
card_insert_of_not_mem (not_mem_empty _), card_empty]
|
||||
... ≤ card S : card_le_card_of_subset H1
|
||||
... = 1 : H,
|
||||
show false, from dec_trivial this)
|
||||
|
||||
proposition eq_singleton_of_card_eq_one {A : Type} {s : set A} {x : A} (H : card s = 1) (xs : x ∈ s) :
|
||||
s = '{x} :=
|
||||
eq_singleton_of_forall_eq xs (take y, assume ys, eq.symm (eq_of_card_eq_one H xs ys))
|
||||
|
||||
proposition exists_eq_singleton_of_card_eq_one {A : Type} {s : set A} (H : card s = 1) : ∃ x, s = '{x} :=
|
||||
have s ≠ ∅, from ne_empty_of_card_pos (by rewrite H; apply dec_trivial),
|
||||
obtain (x : A) (xs : x ∈ s), from exists_mem_of_ne_empty this,
|
||||
exists.intro x (eq_singleton_of_card_eq_one H xs)
|
||||
|
||||
end nat
|
||||
|
||||
-- move to data.set.classical_inverse (and rename file to "inverse")
|
||||
|
||||
theorem inv_fun_spec {X Y : Type} {f : X → Y} {a : set X} {dflt : X} {x : X} (xa : x ∈ a) :
|
||||
f (inv_fun f a dflt (f x)) = f x :=
|
||||
and.right (inv_fun_pos (exists.intro x (and.intro xa rfl)))
|
||||
|
||||
theorem inv_fun_spec' {X Y : Type} {f : X → Y} {a : set X} {dflt : X} {x : X} (xa : x ∈ a) :
|
||||
inv_fun f a dflt (f x) ∈ a :=
|
||||
and.left (inv_fun_pos (exists.intro x (and.intro xa rfl)))
|
||||
|
||||
end set
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -3,7 +3,8 @@ theories
|
|||
|
||||
* [number_theory](number_theory/number_theory.md)
|
||||
* [combinatorics](combinatorics/combinatorics.md)
|
||||
* [finite_group_theory](group_theory/group_theory.md) : group theory based on finsets
|
||||
* [group_theory](group_theory/group_theory.md)
|
||||
* [finite_group_theory](finite_group_theory/finite_group_theory.md) : group theory based on finsets
|
||||
* [topology](topology/topology.md)
|
||||
* [analysis](analysis/analysis.md)
|
||||
* [measure_theory](measure_theory/measure_theory.md)
|
||||
Loading…
Reference in a new issue